Chemotaxis–Haptotaxis System

Abstract

Cancer invasion and metastasis are influenced by a plethora of biochemical processes and involve many biochemical mechanisms, among which chemotaxis and haptotaxis are two of the main mechanisms directing the migration of cancer cells Chaplain and Lolas (2005). Evidence has been found that cancer cells release complex enzymes such as the urokinase-type plasminogen activator (uPA), which degrade the surrounding extracellular matrix (ECM), and thereby allow the migration of cells following the concentration gradient of such diffusive enzymes. This process is referred to as chemotaxis Chaplain and Lolas (2006).

3.1 Introduction

Cancer invasion and metastasis are influenced by a plethora of biochemical processes and involve many biochemical mechanisms, among which chemotaxis and haptotaxis are two of the main mechanisms directing the migration of cancer cells Chaplain and Lolas (2005). Evidence has been found that cancer cells release complex enzymes such as the urokinase-type plasminogen activator (uPA), which degrade the surrounding extracellular matrix (ECM), and thereby allow the migration of cells following the concentration gradient of such diffusive enzymes. This process is referred to as chemotaxis Chaplain and Lolas (2006). On the other hand, in addition to random diffusion, the movement of cancer cells is biased toward the gradient of an immovable stimulus (density of tissue fiber) by finding matrix molecules such as vitronectin adhered therein. This process is called haptotaxis Perumpanani and Byrne (1999).

Recently, a variety of mathematical models have been proposed for various aspects of cancer invasion and metastasis Aznavoorian et al. (1990); Chaplain and Lolas (2005, 2006); Friedman and Lolas (2005); Gatenby and Gawlinski (1996); Meral et al. (2015); Szymańska et al. (2009). Gatenby and Gawlinski (1996) used reaction–diffusion equations to describe the interaction between the density of normal cells, tumor cells and the concentration of $H^+$-ions produced by the latter. They suggested that cancer cells up-regulate certain mechanisms, which allow for the extrusion of excessive protons and hence acidify the environment. This triggers apoptosis of normal cells and thus allows the neoplastic tissue to extend into the space made available. Later on, Meral et al. (2015) proposed a population-based micro–macro model for acid-mediated tumor invasion, which involves the the microscopic dynamics of intracellular protons and their exchange with extracellular counterparts. The continuum micro–macro models explicitly accounting for subcellular events are rather new, especially in the context of cancer cell migration Stinner et al. (2014, 2016).

The analytical results on various models of cancer invasion are mathematically interesting Bellomo et al. (2015); Engwer et al. (2017); Jin (2018); Li and Lankeit (2016); Liţcanu and Morales-Rodrigo (2010b); Morales-Rodrigo and Tello (2014); Stinner et al. (2014, 2016); Szymańska et al. (2009); Tao and Wang (2008); Walker and Webb (2007); Zhigun et al. (2016). From a mathematical point of view, the system under consideration comprises a strong coupling of reaction–diffusion equations and an ordinary differential equation (ODE) in two or three space dimensions. Since ODE corresponds to an everywhere degenerate reaction–diffusion equation and has no regularizing effect, this amounts to considerable difficulty for the analysis. Indeed, analytical results on the cancer invasion model are yet quite fragmentary, so far mainly concentrating on the global existence and boundedness of solutions. For example, Stinner et al. (2014) proved the global existence of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. Very recently, Engwer et al. (2017) studied the global existence of weak solutions to a multiscale model for tumor cell migration in a tissue network. The more detailed answers have been given only in some special cases Hillen et al. (2013); Liţcanu and Morales-Rodrigo (2010b); Tao and Wang (2009); Wang and Ke (2016).

The first part of this chapter is concerned with the Chaplain–Lolas model of cancer invasion Chaplain and Lolas (2005, 2006)

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\varDelta u-\chi \nabla \cdot (u\nabla v)-\xi \nabla \cdot (u\nabla w)+\mu u(r-u-w),\, x\in \varOmega , t>0,\\&\displaystyle {\sigma v_t=\varDelta v- v+u},\quad x\in \varOmega , t>0,\\&\displaystyle {w_t=- vw+\eta w(1-w-u)},\quad x\in \varOmega , t>0,\\&\displaystyle {\frac{\partial u}{\partial \nu }-\chi u\frac{\partial v}{\partial \nu }-\xi u \frac{\partial w}{\partial \nu }=\frac{\partial v}{\partial \nu }=0},\quad x\in \partial \varOmega , t>0,\\&\displaystyle {u(x,0)=u_0(x)}, \sigma v(x,0)= \sigma v_0(x),w(x,0)=w_0(x),\quad x\in \varOmega \\ \end{aligned}\right. \end{aligned}$$

(3.1.1)

in a bounded domain $\varOmega \subset \mathbb {R}^n$ $(n=2,3)$ with smooth boundary $\partial \varOmega$, where $\partial /\partial \nu$ denotes the outward normal derivative on $\partial \varOmega$, u denotes the density of cancer cells, v represents the concentration of the matrix degrading enzyme (MDE) and w describes the concentration of the extracellular matrix (ECM), respectively; and $\chi$ and $\xi$ measure the chemotactic and haptotactic sensitivities, respectively. The term $\mu u(r-u-w)$ assumes that in the absence of the ECM, cancer cell proliferation satisfies a logistic law, and $\eta >0$ embodies the ability of the ECM to remodel back to a normal level. The parameter $\sigma$ may take on the value of 0 or 1. When $\sigma =0$, we are making the simplifying assumption that the diffusion rate of the MDE is much greater than that of cancer cells, which is supported by evidence Chaplain and Lolas (2006). Indeed, similar quasi-steady approximations for corresponding chemoattractant equations are frequently used to study classical chemotaxis systems (see Jäger and Luckhaus (1992)). As for the initial data $(u_0,v_0, w_0)$, we suppose throughout this section that, for some $\vartheta \in (0,1)$,

$$\begin{aligned} \left\{ \begin{aligned}&\displaystyle {u_0\in C^1(\bar{\varOmega })~~\text{ with }~~u_0\ge 0~~\text{ in }~~\varOmega ,~~u_0\not \equiv 0}, \\&\displaystyle {v_0\in W^{1,\infty }(\varOmega )~~\text{ with }~~v_0\ge 0~~\text{ in }~~\varOmega ,~}\\&\displaystyle {w_0\in C^{2+\vartheta }(\bar{\varOmega })~~\text{ with }~~w_0\ge 0~~\text{ in }~~\bar{\varOmega }~~\text{ and }~~\frac{\partial w_0}{\partial \nu }=0~~\text{ on }~~\partial \varOmega .} \\ \end{aligned} \right. \end{aligned}$$

(3.1.2)

It is observed that when letting $w\equiv 0$, (3.1.1) is reduced to the Keller–Segel system with logistic source. This chemotaxis-only system has been extensively studied by many authors during the last decades. In this context, the particular attention focuses on the question of whether the solutions of the models are bounded or blow-up (see, e.g., Cieślak and Stinner (2012), Cieślak and Winkler (2008), Ishida et al. (2014), Painter and Hillen (2002) and Winkler (2008, 2010a, 2011b)). In particular, solutions may blow up in finite time when $n\ge 2, \mu = 0$ Herrero and Velázquez (1997); Nagai (2001). It is known that arbitrarily small $\mu > 0$ guarantees the boundedness of solutions when $n= 2$ Osaki et al. (2002), while when $n \ge 3$, appropriately large $\mu$ (compared with the chemotactic coefficient $\chi$) is required to exclude unbounded solution Winkler (2010a). It is still unknown whether finite-time blow-up may occur if $\mu >0$ is small, though global weak solutions are known to exist and will become smooth after some time Lankeit (2015). On the other hand, the nonlinear self-diffusion of cells may prevent blow-up of solutions Cieślak and Winkler (2008); Ishida et al. (2014); Wang et al. (2014).

When $\chi = 0$, (3.1.1) becomes the haptotaxis-only system. For $\chi =\mu =\eta = 0, \sigma =1$, the local existence and uniqueness of classical solutions have been shown in Morales-Rodrigo (2008). The global existence and asymptotic behavior of weak solutions have been proven in Liţcanu and Morales-Rodrigo (2010b); Marciniak-Czochra and Ptashnyk (2010); Walker and Webb (2007) when $\eta =0$, and global existence and uniqueness of classical solutions have been shown in Tao (2011) when $\eta > 0$, respectively.

Note that in contrast of the chemotaxis-only system, haptotaxis-only system and the chemotaxis–haptotaxis system, the chemotaxis–haptotaxis system with remodeling of non-diffusible attractant ($\eta >0$ in (3.1.1)) is much less understood (Chaplain and Lolas (2006), Pang and Wang (2017) and Tao and Winkler (2014b)). The main technical difficulty in their proof lies in the effects of the strong coupling in (3.1.1) on the spatial regularity of u,  v and w when $\eta >0.$ When $\eta =0,$ one can build a one-sided pointwise estimate which connects $\varDelta w$ to v (see Lemma 2.2 of Cao (2016) or (3.10) of Wang (2016)). Relying on such a pointwise estimate, one can derive two useful energy-type inequalities which can help us to bypass the term $\int _{\varOmega } u^{p-1}\nabla \cdot ( u \nabla w)$ (see Lemma 3.2 of Zheng (2017b)). Using such information along with coupled estimate techniques and the boundedness of the $\Vert \nabla v(\cdot , t)\Vert _{L^2(\varOmega )}$, one can establish the estimates on $\int _{\varOmega } u^p+|\nabla v|^{2q}$ for any p and $q > 1$ (see Lemmas 3.3 and 3.4 of Zheng (2017b)), which combined with the standard regularity theory of parabolic equation and the Moser iteration procedure implies the boundedness of u in $L^\infty (\varOmega )$ (see Lemma 3.5 of Zheng (2017b)). However, for the model (3.1.1) with $\eta >0$, one needs to estimate the chemotaxis-related integral term $\int _{\varOmega }a^p|\nabla v|^2$ (see (3.28) in Tao and Winkler (2014b)) or $\int _{\varOmega }e^{-(p+1)(t-s)}a^p|\nabla v|^2$ (see (3.8) of Pang and Wang (2017)) with $a:= ue^{-\xi w}$, which requires much more technical demanding. In Pang and Wang (2017), assuming that $\mu >\xi \eta \max \{\Vert u_0\Vert _{L^\infty (\varOmega )},1\}+\mu ^*(\chi ^2,\xi )$ (the hypothesis cannot be dropped (see the proof of Lemma 3.2 of Pang and Wang (2017))), Pang and Wang (2017) proved that the problem 3.1.1 admits a unique global solution $(u,v,w)\in (C^{2,1} (\bar{\varOmega }\times (0,\infty )))^3$. Moreover, u is bounded in $\varOmega \times (0,\infty )$.

This chapter consists of three parts. The first part shows the global boundedness of classical solutions to the chemotaxis–haptotaxis model with any $\eta >0$ (Ke and Zheng (2018)).

Theorem 3.1

Let $\sigma>0,\chi>0, \xi >0, r=1$ and $\eta > 0$. Assume that $\varOmega \subseteq \mathbb {R}^2$ is a bounded domain with smooth boundary and the initial data $(u_0, v_0,w_0)$ satisfy

$$\begin{aligned} \left\{ \begin{aligned}&\displaystyle {u_0\in C^{2+\vartheta }(\bar{\varOmega })~~\text{ with }~~u_0\ge 0~~\text{ in }~~\varOmega ~~\text{ and }~~\frac{\partial u_0}{\partial \nu }=0~~\text{ on }~~\partial \varOmega },\\&\displaystyle {v_0\in C^{2+\vartheta }(\bar{\varOmega })~~\text{ with }~~v_0\ge 0~~\text{ in }~~\varOmega ~~\text{ and }~~\frac{\partial v_0}{\partial \nu }=0~~\text{ on }~~\partial \varOmega },\\&\displaystyle {w_0\in C^{2+\vartheta }(\bar{\varOmega })~~\text{ with }~~w_0\ge 0~~\text{ in }~~\bar{\varOmega }~~\text{ and }~~\frac{\partial w_0}{\partial \nu }=0~~\text{ on }~~\partial \varOmega } \\ \end{aligned} \right. \end{aligned}$$

with some $\vartheta \in (0,1).$ If $\mu >0$, then there exists a triple $(u,v,w)\in (C^{0} (\bar{\varOmega }\times [0,\infty ))\times C^{2,1} (\bar{\varOmega }\times (0,\infty )))^3$ which solves (3.1.1) in the classical sense. Moreover, u and v are bounded in $\varOmega \times (0,\infty )$.

Remark 3.1

(i) If $w\equiv 0$, it is not difficult to obtain that the solutions under the conditions of Theorem 3.1 are uniformly bounded when $n=2$, which coincides with the results of Osaki et al. (2002).

(ii) From Theorem 3.1, it follows that solutions of model (3.1.1) are global and bounded for any $\eta =0,\mu >0$ and $n\le 2$, which coincides with the result of Tao (2014).

The second part of this chapter is devoted to the integrative interactions of chemotaxis, haptotaxis, logistic growth and remodeling mechanisms, and establishes the global existence of classical solutions to the chemotaxis–haptotaxis model (3.1.1) with the remodeling of the ECM. It is noticed that the authors of Tao and Winkler (2014b) made appropriate use of the dampening effect of $-\eta uw$ in the third equation of (3.1.1) to derive an energy-like inequality, which yields an a priori estimate of $\int _\varOmega u\ln u$ in bounded time intervals. The latter is the starting point for a bootstrap argument used to derive higher regularity estimates. In this part, thanks to the variable transformation $a = ue^{-\xi w}$ (Tao and Wang (2009, 2008); Tao and Winkler (2014b)), making use of the damping effect in the first equation of cancer cells, one derives a priori estimate of $\int _\varOmega u\ln u$ for all time $t>0$ and thus proves the global boundedness of solutions thereof rather comprehensively. The result in this respect is the following (Pang and Wang (2018)).

Theorem 3.2

Let $\varOmega \subset \mathbb {R}^2$ be a bounded smooth domain, and suppose that $\chi>0, \xi >0$, $\eta >0$ and $\mu >0$. Then for any $r>0$, the problem (3.1.1) admits a unique global classical solution (u, v, w), where $\Vert u(\cdot ,t)\Vert _{L^\infty {(\varOmega )}}$ is uniformly bounded for $t\in (0,\infty )$.

The key step of our analysis of (3.1.1) consists of identifying a certain dissipative property of the functional $\int _\varOmega e^{\xi w}a^2$ with $a=e^{-\xi w}u$. Indeed, we shall see in Lemma 3.17 below that a certain variant thereof satisfies an inequality of the form

$$ \displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2 +\frac{1}{\varepsilon }\int _\varOmega e^{\xi w}a^2 \le c (\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert a\Vert _{L^2(\varOmega )}^2)\displaystyle \int _\varOmega e^{\xi w}a^2+c(\varepsilon ) $$

with some $c>0, c(\varepsilon )>0$ for any $\varepsilon >0$ (see (3.4.17)), whereupon Lemma 3.5 will provide the bound of $\int _\varOmega u^2$, which provides a starting point for the higher regularity estimates of solutions. On the other hand, in the case of $\sigma =0$, the key step in our proof of theorem is to identify

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a\ln a +\frac{\mu }{2} \int _\varOmega e^{\xi w}a\ln a\\ \le&\varepsilon c (\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)\displaystyle \int _\varOmega e^{\xi w}a\ln a \\&+c( \Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)+c(\varepsilon ) \end{aligned} \end{aligned}$$

(3.1.3)

with some $c>0, c(\varepsilon )>0$ for all $\varepsilon >0$, which along with Lemma 3.5 enables us to obtain an a priori estimate for u in the space $L\log L(\varOmega )$ for all time. Notice that in the two-dimensional space, the global boundedness of solutions (p, c, w) to a tumor angiogenesis model was established in Morales-Rodrigo and Tello (2014) if the initial data $w_0$ of the fibronectin concentration satisfies either $w_0>1$ or $\Vert w_0-1\Vert _{L^\infty (\varOmega )}<\delta$ with some $\delta >0$. It should be mentioned that by the estimate technique above, one can remove the extra assumption on $w_0$.

In the three-dimensional setting, the problem of the global existence of solutions to (3.1.1) seems to be more delicate. Indeed, the only result that we are aware of is presented in the recent paper Bellomo et al. (2015), where a certain global weak solution was constructed for (3.1.1) with $\sigma =1$. To the best of our knowledge, the existence of global classical solutions to (3.1.1) is still open. As mentioned previously, some weak solutions to the three-dimensional chemotaxis system including logistic growth eventually become classical solutions after some waiting time when smallness conditions on the growth rate of the cells are imposed Lankeit (2015); Winkler (2008). A natural question is whether the chemotaxis–haptotaxis system (3.1.1) possesses global classical solutions under some smallness conditions. Our result in this direction is as follows (Pang and Wang (2018)).

Theorem 3.3

Let $\varOmega \subset \mathbb {R}^3$ be a bounded convex domain with smooth boundary and $\chi>0, \xi >0$, $\eta >0$ and $\mu >0$. For any given $w_0$, there exist a constant $r_0 = r_0(\mu , |\varOmega |, \Vert w_0\Vert _{L^\infty (\varOmega )}) > 0$ and appropriate small $\Vert u_0\Vert _{L^2(\varOmega )}$ and $\Vert v_0\Vert _{W^{1,4}(\varOmega )}$ such that the problem (3.1.1) possesses a unique global classical solution (u, v, w) provided that $r< r_0$.

Relying on the mass evolution of solutions to (3.1.1), the quantity $\int _\varOmega a^2 (t)+\int _\varOmega |\nabla v(t)|^4$ is shown to satisfy an autonomous ordinary differential inequality, and thereby is bounded whenever $r>0$ and initial data are suitably small by the comparison argument of the corresponding ordinary differential equation. This serves as a starting point for the bootstrap procedure to yield a bound for a in $L^\infty (\varOmega )$.

As a physiological process, angiogenesis involves the formation of new capillary networks sprouting from a pre-existing vascular network and plays an important role in embryo development, wound healing and tumor growth. For example, it has been recognized that capillary growth through angiogenesis leads to the vascularization of a tumor, providing it with its own dedicated blood supply and consequently allowing for rapid growth and metastasis.

The process of tumor angiogenesis can be divided into three main stages (which may be overlapping): (i) changes within existing blood vessels; (ii) formation of new vessels; and (iii) maturation of new vessels. Over the past decade, a lot of work has been done on the mathematical modeling of tumor growth; see, for example, Anderson and Chaplain (1998b); Bellomo et al. (2015); Chaplain and Lolas (2005, 2006); Li et al. (2015); Stinner et al. (2015, 2016) and the references cited therein. In particular, the role of angiogenesis in tumor growth has also attracted a great deal of attention; see, for example, Anderson and Chaplain (1998a); Chaplain and Stuart (1993); Levine et al. (2001); Paweletz and Knierim (1989); Sleeman (1997) and the references cited therein. For example, in Levine et al. (2001), a system of PDEs using reinforced random walks was deployed to model the first stage of angiogenesis, in which chemotactic substances from the tumor combine with the receptors on the endothelial cell wall to release proteolytic enzymes that can degrade the basal membrane of the blood vessels eventually.

The third part of this chapter considers a variation of the model proposed in Anderson and Chaplain (1998b), namely

$$\begin{aligned} \left\{ \begin{aligned}&p_t=\varDelta p-\nabla {\cdot } p(\displaystyle \frac{\alpha }{1+c}\nabla c+\rho \nabla w)+\lambda p(1-p),\,&x\in \varOmega , t>0,\\&c_t=\varDelta c-c-\mu pc,\,&x\in \varOmega , t>0,\\&w_t= \gamma p(1-w),\,&x\in \varOmega , t>0, \\&\frac{\partial p}{\partial \nu }- p(\displaystyle \frac{\alpha }{1+c}\frac{\partial c}{\partial \nu }+ \rho \frac{\partial w}{\partial \nu } )=\frac{\partial c}{\partial \nu }=0,&x\in \partial \varOmega , t>0,\\&p(x,0)=p_0(x), \ c(x,0)=c_0(x), \ w(x,0)= w_0(x),&x\in \varOmega ,\\ \end{aligned}\right. \end{aligned}$$

(3.1.4)

in a bounded smooth domain $\varOmega \subset \mathbb {R}^N (N=1,2)$, where, in addition to random motion, the existing blood vessels’ endothelial cells p migrate in response to the concentration gradient of a chemical signal c (called Tumor Angiogenic Factor, or TAF) secreted by tumor cells as well as the concentration gradient of non-diffusible glycoprotein fibronectin w produced by the endothelial cells Morales-Rodrigo and Tello (2014). The formerly directed migration is a chemotactic process, whereas the latter is a haptotactic process. In this model, it is assumed that the endothelial cells proliferate according to a logistic law, that the spatio-temporal evolution of TAF occurs through diffusion, natural decay and degradation upon binding to the endothelial cells, and that the fibronectin is produced by the endothelial cells and degrades upon binding to the endothelial cells.

For the remainder of this chapter, the initial data are assumed to satisfy

$$\begin{aligned} \left\{ \begin{aligned}&(p_0,c_0,w_0)\in (C^{2+\beta }(\overline{\varOmega }))^3~\hbox {is nonnegative}~\hbox {for some}~\beta \in (0,1) ~ \hbox {with}~p_0\not \equiv 0,\\[.2cm]&\displaystyle {\frac{\partial p_0}{\partial \nu }- p_0(\displaystyle \frac{\alpha }{1+c_0}\frac{\partial c_0}{\partial \nu }+ \rho \frac{\partial w_0}{\partial \nu })=\frac{\partial c_0}{\partial \nu }=0}. \end{aligned} \right. \end{aligned}$$

(3.1.5)

The third part focuses on the global existence and asymptotic behavior of classical solutions to (3.1.4). Let us look at two subsystems contained in (3.1.4). The first is a Keller–Segel-type chemotaxis system with signal absorption:

$$\begin{aligned} \left\{ \begin{aligned}&p_t=\varDelta p-\nabla \cdot ( p\nabla c)+\lambda p(1-p),\quad&x\in \varOmega , t>0, \\&c_t=\varDelta c- pc,\quad&x\in \varOmega , t>0. \\ \end{aligned}\right. \end{aligned}$$

(3.1.6)

It is known that, unlike the standard Keller–Segel model, (3.1.6) with $\lambda =0$ possesses global, bounded classical solutions in two-dimensional bounded convex domains for arbitrarily large initial data; while in three spatial dimensions, it admits at least global weak solutions which eventually become smooth and bounded after some waiting time Tao and Winkler (2012c). In the high-dimensional setting, it has been proved that global bounded classical solutions exist for suitably large $\lambda >0$, while only certain weak solutions are known to exist for arbitrary $\lambda >0$ Lankeit and Wang (2017).

Another delicate subsystem of (3.1.4) is the haptotaxis-only system obtained by letting $\alpha =0$ in (3.1.4):

$$\begin{aligned} \left\{ \begin{aligned}&p_t=\varDelta p-\rho \nabla {\cdot }(p\nabla w)+\lambda p(1-p),\,&x\in \varOmega , t>0,\\&w_t= \gamma p(1-w),\,&x\in \varOmega , t>0. \\ \end{aligned}\right. \end{aligned}$$

Here, since the quantity w satisfies an ODE without any diffusion, the smoothing effect on the spatial regularity of w during evolution cannot be expected. To the best of our knowledge, unlike the study of chemotaxis systems, the mathematical literature on haptotaxis systems is comparatively thin. Indeed, the literature provides only some results on global solvability in various special models, and the detailed description of qualitative properties such as long-time behaviors of solutions is available only in very particular cases (see, for example, Corrias et al. (2004); Liţcanu and Morales-Rodrigo (2010b, 2010a); Marciniak-Czochra and Ptashnyk (2010); Tao (2011); Tao and Winkler (2019a); Walker and Webb (2007); Winkler (2018b)).

More recently, some results on global existence and asymptotic behavior for certain chemotaxis–haptotaxis models of cancer invasion have been obtained (see, for example, Li and Lankeit (2016); Pang and Wang (2017, 2018); Stinner et al. (2014); Tao and Winkler (2014b, 2015a); Wang and Ke (2016)). Particularly, Hillen et al. (2013) have shown the convergence of a cancer invasion model in one-dimensional domains, and the result has been subsequently extended to higher dimensions Li and Lankeit (2016); Tao and Winkler (2015a); Wang and Ke (2016).

In Morales-Rodrigo and Tello (2014), in two spatial dimensions, the authors showed the global existence and long-time behavior of classical solutions to (3.1.4) when the initial data $(p_0, c_0, w_0)$ satisfy either $w_0 > 1$ or $\Vert w_0-1\Vert _{L^\infty (\varOmega )}<\delta$ for some $\delta > 0$ (see Lemma 5.8 of Morales-Rodrigo and Tello (2014)). Generalizing this result, our first main result establishes that, for any choice of reasonably regular initial data $(p_0, c_0, w_0)$, the $L^\infty$-norm of p is globally bounded. This is done via an iterative method (Pang and Wang (2019)).

Theorem 3.4

Let $\alpha , \rho , \lambda , \mu$ and $\gamma$ be positive parameters. Then for any initial data $(p_0, c_0, w_0)$ satisfying (3.1.5), the problem (3.1.4) possesses a unique classical solution (p, c, w) comprising nonnegative functions in $C(\bar{\varOmega }\times [0,\infty ))\cap C^{2,1}(\bar{\varOmega }\times (0,\infty )$ such that $\Vert p(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C$ for all $t>0$.

Next, we investigate the asymptotic behavior of solutions to (3.1.4). Under an additional mild condition on the initial data $w_0$, we will show that the solution (p, c, w) converges to the spatially homogeneous equilibrium (1, 0, 1) as time tends to infinity (Pang and Wang (2019)).

Theorem 3.5

Let $\alpha , \rho , \lambda , \mu$ and $\gamma$ be positive parameters, and suppose that (3.1.5) is satisfied and $w_0>1-\displaystyle \frac{1}{\rho }$. Then the solution $(p,c,w)\in C(\bar{\varOmega }\times [0,\infty ))\cap C^{2,1}(\bar{\varOmega }\times (0,\infty )$ of (3.1.4) satisfies

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert p(\cdot ,t)-1\Vert _{L^r(\varOmega )}+\Vert c(\cdot ,t)\Vert _{W^{1,2}(\varOmega )}+\Vert w(\cdot ,t)-1\Vert _{L^r(\varOmega )}=0 \end{aligned}$$

(3.1.7)

for any $r\ge 2$. In particular, if $N=1$, then for any $\epsilon \in (0,\min \{\lambda _1,1,\gamma ,\lambda \})$ there exists $C(\epsilon )>0$ such that

$$\begin{aligned} \displaystyle \Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )} \le C(\epsilon )e^{-(\min \{\lambda _1,1,\gamma ,\lambda \}-\epsilon ) t}, \end{aligned}$$

(3.1.8)

$$\begin{aligned} \Vert c(\cdot ,t)\Vert _{W^{1,2}(\varOmega )}\le C(\epsilon )e^{-(1-\epsilon )t}, \end{aligned}$$

(3.1.9)

$$\begin{aligned} \displaystyle \Vert w(\cdot ,t)-1\Vert _{W^{1,2}(\varOmega )} \le C(\epsilon )e^{-(\gamma -\epsilon ) t}, \end{aligned}$$

(3.1.10)

where $\lambda _1>0$ is the first nonzero eigenvalue of $-\varDelta$ in $\varOmega$ with the homogeneous Neumann boundary condition.

The main mathematical challenge of the full chemotaxis–haptotaxis system is the strong coupling between the migratory cells p and the haptotactic agent w. This strong coupling has an important effect on the spatial regularity of p and w. In fact, the lack of regularization effect in the spatial variable in the w-equation and the presence of p therein demand tedious estimates on the solution. The key ideas behind this result are as follows.

As pointed out in Tao and Winkler (2015a), the variable transformation $z:= p e^{-\rho w}$ plays an important role in the examination of global solvability for the full chemotaxis–haptotaxis model in the two- and higher dimensional setting. However, due to the presence of the additional chemotaxis term in our model, this approach is not directly applicable to our problem. Instead, in the derivation of Theorem 3.4, we introduce the variable transformation $q := p(c + 1)^{-\alpha } e^{-\rho w}$ as in Morales-Rodrigo and Tello (2014), and thereby ensure that $q(\cdot ,t)$ is bounded in $L^ n(\varOmega )$ for any finite n (see Lemma 3.27). It is essential to our approach to derive a bound for $\int _{\varOmega }q^{2^{m+1}}+\int ^{t+\tau }_t\int _{\varOmega } |\nabla q^{2^{m}} |^2$ from the bound of $\int ^{t+\tau }_t\int _{\varOmega }q^{2^{m}}$ ($m=1,2,\ldots$) by making appropriate use of (3.5.3)–(3.5.4) in Lemma 3.26 (see (3.5.7) below).

3.2 Preliminaries

Before formulating our main results, we recall some preliminary lemmas used throughout this chapter. Some basic properties of solution can be found in Horstmann and Winkler (2005) (see also Winkler (2010), Zhang and Li (2015b)).

Lemma 3.1

(Horstmann and Winkler (2005)) For $p\in (1,\infty )$, let $A := A_p$ denote the sectorial operator defined by

$$ A_pu :=-\varDelta u~~\text{ for } \text{ all }~~u\in D(A_p) :=\{\varphi \in W^{2,p}(\varOmega )|\frac{\partial \varphi }{\partial \nu }|_{\partial \varOmega }=0\}. $$

The operator $A + 1$ possesses fractional powers $(A + 1)^{\alpha }(\alpha \ge 0)$, the domains of which have the embedding properties

$$ D((A+1)^\alpha )\hookrightarrow W^{1,p}(\varOmega )~~\text{ if }~~\alpha >\frac{1}{2}. $$

If $m\in \{0, 1\}$, $p\in [1,\infty ]$ and $q \in (1,\infty )$ with $m-\frac{n}{p} < 2\alpha -\frac{n}{q}$, then we have

$$ \Vert u\Vert _{W^{m,p}(\varOmega )}\le C\Vert (A+1)^\alpha u\Vert _{L^{q}(\varOmega )}~~~\text{ for } \text{ all }~~u\in D((A+1)^\alpha ), $$

where C is a positive constant. The fact that the spectrum of A is a p-independent countable set of positive real numbers $0 = \kappa _0< \kappa _1<\kappa _2 <\cdots$ entails the following consequences: for all $1\le p< q < \infty$ and $u\in L^p(\varOmega )$, it has

$$ \Vert (A+1)^\alpha e^{-tA}u\Vert _{L^q(\varOmega )}\le ct^{-\alpha -\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}e^{(1-\kappa )t}\Vert u\Vert _{L^p(\varOmega )} $$

for any $t > 0$ and $\alpha \ge 0$ with some $\kappa > 0.$

In deriving some preliminary estimates for v, we shall make use of the following property referred to as a variation of Maximal Sobolev Regularity.

Lemma 3.2

([Hieber and Prüss 1997, Theorem 3.1] [Li and Wang 2018, Lemma 2.2]) Let $r\in \left( 1,\infty \right)$ and $\kappa >0$; consider the following evolution equation:

$$\begin{aligned} \left\{ \begin{aligned}&h_{t}=\varDelta h-\kappa h+f,\quad \quad&\left( x,t \right) \in \varOmega \times \left( 0,T \right) ,\\&\nabla h\cdot \nu =0,&\left( x,t \right) \in \partial \varOmega \times \left( 0,T \right) ,\\&h\left( x,0 \right) =h_{0}(x),&x\in \varOmega .\\ \end{aligned} \right. \end{aligned}$$

(3.2.1)

Then for each $h_{0}\in W^{2,r}\left( \varOmega \right)$ with $\nabla h_{0}\cdot \nu =0$ on $\partial \varOmega$ and any $f\in L^{r}\left( \left( 0,T \right) ,L^{r} \left( \varOmega \right) \right)$, (3.2.1) admits a unique mild solution $h\in W^{1,r}\left( \left( 0,T \right) ;L^{r} \left( \varOmega \right) \right) \cap L^{r}\left( \left( 0,T \right) ;W^{2,r}\left( \varOmega \right) \right)$. Moreover, for any $\varepsilon \in (0,\kappa ]$, there exists $C_r>0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{0}^Te^{\varepsilon r s}\Vert h(\cdot ,s)\Vert ^r_{W^{2,r}(\varOmega )}ds\le C_r\left( \int _{0}^Te^{\varepsilon r s} \Vert f(\cdot ,s)\Vert ^r_{L^r(\varOmega )}ds+(\Vert h_0(\cdot )\Vert ^r_{W^{2,r}(\varOmega )})\right) .}\\ \end{aligned} \end{aligned}$$

(3.2.2)

Let us also recall the well-known Gagliardo–Nirenberg inequality Friedman (1969); Tao and Wang (2009).

Lemma 3.3

Let $\varOmega \subset \mathbb {R}^n$ be a bounded domain with smooth boundary. Let l, k be any integers satisfying $0 \le l<k$, $1\le q, r \le \infty$ and $p\in \mathbb {R}^+$, $\frac{l}{k} \le \theta \le 1$ such that

$$\begin{aligned} \frac{1}{p}- \frac{l}{n}=\theta (\frac{1}{q}- \frac{k}{n})+\frac{1-\theta }{r}. \end{aligned}$$

(3.2.3)

Then there are positive constants $C_{GN}$ and $C_1$ depending only on $\varOmega ,q,k,r$ and n such that for any function $\varphi \in W^{k,q}(\varOmega )\cap L^r(\varOmega )$,

$$\begin{aligned} \Vert \nabla ^l \varphi \Vert _{L^p(\varOmega )} \le C_{GN}\Vert \nabla ^k \varphi \Vert _{L^q(\varOmega )}^{\theta }\Vert \varphi \Vert ^{1-\theta }_{L^r(\varOmega )}+C_1\Vert \varphi \Vert _{L^r(\varOmega )} \end{aligned}$$

(3.2.4)

with the following exception: If $1<q<\infty$ and $k-l- \frac{n}{q}$ is a nonnegative integer, then we assume that (3.2.3) holds for $\theta$ satisfying $\frac{l}{k} \le \theta < 1, r>1$.

To estimate $\int _\varOmega a^2 +\int _\varOmega |\nabla v|^4$ with $a=e^{-\xi w}u$ in the proof Theorem 3.3, we will have to get a handle on $\int _\varOmega |\nabla v|^6$ and $\int _\varOmega a^3$. The above Gagliardo–Nirenberg inequality enables us to replace them by more convenient terms.

Lemma 3.4

Let $\varOmega \subset \mathbb {R}^3$ be a bounded domain with smooth boundary. For any $\varepsilon >0$, there are $C(\varepsilon )>0$ and $C_2>0$ such that for any $v\in C^2(\varOmega )$

$$\begin{aligned} \int _\varOmega |\nabla v|^6\le \varepsilon \int _\varOmega |\nabla |\nabla v|^2|^2+C(\varepsilon )\left( \left( \int _\varOmega |\nabla v|^4\right) ^3+ \left( \int _\varOmega |\nabla v|^4\right) ^{\frac{3}{2}}\right) , \end{aligned}$$

(3.2.5)

and for any $a\in W^{1,2}(\varOmega )$

$$\begin{aligned} \int _\varOmega a^3\le \varepsilon \int _\varOmega |\nabla a|^2+C(\varepsilon ) \left( \int _\varOmega a^2\right) ^3+ C_2 \left( \int _\varOmega a\right) ^3. \end{aligned}$$

(3.2.6)

Proof

We would like refer the reader to Lemma 4.3 of Lankeit (2015) for (3.2.5) and (2.7) with $\gamma =2$ of Winkler (2008) for (3.2.6), respectively.

The following statement generalizing that of Lemma 3.4 in Stinner et al. (2014) plays an important role in the proofs of Lemmas 3.17 and 3.22 below.

Lemma 3.5

Let $T\in (0,\infty ],0<\tau <T$ and suppose that y is a nonnegative absolutely continuous function satisfying

$$\begin{aligned} y'(t)+a(t)y(t)\le b(t)y(t)+c(t) \quad ~~~~~\hbox {for a.e. }t\in (0,T) \end{aligned}$$

(3.2.7)

with some functions $a(t)>0,b(t)\ge 0, c(t)\ge 0$ and $a,b,c\in L^1_{loc}(0,T)$ for which there exist $b_1,c_1>0$ and $\rho >0$ such that

$$ \displaystyle \sup _{0\le t\le T-\tau }\int ^{t+\tau }_t b(s)ds\le b_1,~~ \displaystyle \sup _{0\le t\le T-\tau }\int ^{t+\tau }_t c(s)ds\le c_1 $$

and

$$ \int ^{t+\tau }_t a(s)ds-\int ^{t+\tau }_t b(s)ds\ge \rho ~~~\hbox {for any}~~~ t\in (0,T-\tau ).$$

Then

$$ y(t)\le y(0)e^{b_1}+\displaystyle \frac{c_1 e^{2b_1}}{1-e^{-\rho }}+c_1e^{b_1}~~\hbox { for all}~~ t\in (0,T). $$

Proof

From (3.2.7) and a comparison argument, we obtain that for any $\tau \le t<T$,

$$\begin{aligned} \begin{aligned} y(t)\le&y(t-\tau )e^{\int ^t_{t-\tau }(b(s)-a(s))ds} +\displaystyle \int ^t_{t-\tau }c(s)e^{\int ^t_s(b(\sigma )-a(\sigma ))d\sigma } ds\\ \le&y(t-\tau ) e^{-\rho }+\displaystyle \int ^t_{t-\tau }c(s)e^{\int ^t_sb(\sigma )d\sigma }ds\\ \le&y(t-\tau ) e^{-\rho }+\displaystyle \int ^t_{t-\tau }c(s)e^{\int ^t_{t-\tau }b(\sigma )d\sigma }ds\\ \le&y(t-\tau ) e^{-\rho }+c_1e^{b_1}. \end{aligned} \end{aligned}$$

Hence, taking $t=k\tau$ $(k=1,2,\ldots ,[\frac{T}{\tau }])$, we have

$$ \begin{aligned} y(k\tau )\le&y((k-1)\tau )e^{-\rho }+c_1e^{b_1}\\ \le&e^{-\rho } (y((k-2)\tau )e^{-\rho }+c_1e^{b_1})+c_1e^{b_1}\\ =&e^{-2\rho }y((k-2)\tau )+e^{-\rho }c_1e^{b_1}+c_1e^{b_1}\\ =&e^{-k\rho }y(0)+ c_1 e^{b_1}\displaystyle \sum _{j=0}^{k-1} e^{-j\rho }\\ \le&e^{-k\rho } y(0)+\displaystyle \frac{c_1 e^{b_1}}{1-e^{-\rho }}. \end{aligned} $$

Now for any given $t\in (0, T)>0$, we can fix $k\in \mathbb {N}$ such that $k\tau < t\le (k+1)\tau$, i.e., $k=[\frac{t}{\tau }]$, and thus get

$$ \begin{aligned} y(t)\le&y(k\tau )e^{\int ^t_{k\tau }(b(s)-a(s))ds} +\displaystyle \int ^t_{k\tau }c(s)e^{\int ^t_s(b(\sigma )-a(\sigma ))d\sigma } ds\\ \le&y(k\tau )e^{\int ^t_{k\tau }b(s)ds}+\displaystyle \int ^t_{k\tau }c(s) e^{\int ^t_sb(\sigma )d\sigma }ds\\ \le&y(k\tau )e^{b_1}+c_1e^{b_1}\\ \le&y(0)e^{b_1}e^{-[\frac{t}{\tau }]\rho }+\displaystyle \frac{c_1 e^{2b_1}}{1-e^{-\rho }}+c_1e^{b_1}. \end{aligned} $$

Apart from the asserted results in Lemma 3.2, we also need some fundamental estimates for the inhomogeneous linear heat equation

$$\begin{aligned} \left\{ \begin{aligned} v_t=\varDelta v -v+u,\quad x\in \varOmega , t>0,\\ \displaystyle \frac{\partial v}{\partial \nu }=0,\quad x\in \partial \varOmega , t>0,\\ v(x,0)=v_0(x),\quad x\in \varOmega ,\\ \end{aligned}\right. \end{aligned}$$

(3.2.8)

which can be derived from a standard regularity argument involving the variation-of-constants formula for v and $L^q-L^p$ estimates for the heat semigroup (see Horstmann and Winkler (2005) for instance).

Lemma 3.6

[Yang et al. 2015, Lemma 2.2] Let $T>0$, $1\le p\le \infty , v_0\in L^p(\varOmega )$ and $u\in L^1(0,T;L^p(\varOmega ))$. Then (3.2.8) has a unique mild solution $v\in C([0,T];L^p(\varOmega ) )$ given by

$$ v(t)=e^{-t}e^{t\varDelta }v_0+\int ^t_0 e^{-(t-s)}e^{(t-s)\varDelta }u(s)ds \quad ~~\hbox {for all}~~ t\in [0,T], $$

where $e^{t\varDelta }$ is the semigroup generated by the Neumann Laplacian. In addition, let $1\le q\le p< \frac{nq}{n-q}$, $v_0\in W^{1,p}(\varOmega )$ and $u\in L^\infty (0,T; L^q(\varOmega ))$. Then for every $t\in (0,T)$,

$$\begin{aligned} \Vert v(t)\Vert _{L^p(\varOmega )}\le \Vert v_0\Vert _{L^p(\varOmega )}+ c_2\Vert u\Vert _{L^\infty ( (0, T);L^q(\varOmega ))}, \end{aligned}$$

(3.2.9)

$$\begin{aligned} \Vert \nabla v(t)\Vert _{L^p(\varOmega )}\le \Vert \nabla v_0\Vert _{L^p(\varOmega )}+ c_2\Vert u\Vert _{L^\infty ( (0, T);L^q(\varOmega ))}, \end{aligned}$$

(3.2.10)

where $c_2$ is a positive constant depending on p, q and n.

The following statement can be found in Appendix A of Tao and Winkler (2014b).

Lemma 3.7

Let $\varOmega \subset \mathbb {R}^2$ be a bounded domain with smooth boundary. Then for all $M>0$, there exist constants $\alpha >0$, $\beta >0$ depending only upon M such that for any nonnegative function $u\in L^2(\varOmega )$ and $\int _\varOmega u\le M$, the solution v of

$$\begin{aligned} \left\{ \begin{aligned} -\varDelta v+v=u,\quad x\in \varOmega ,\\ \displaystyle \frac{\partial v}{\partial \nu }=0,\quad x\in \partial \varOmega \\ \end{aligned}\right. \end{aligned}$$

(3.2.11)

satisfies

$$\begin{aligned} \int _\varOmega | \nabla v|^2+\int _\varOmega |v|^2\le \alpha \int _\varOmega u\ln u+\beta . \end{aligned}$$

(3.2.12)

3.3 Global Boundedness of Solutions to a Chemotaxis–Haptotaxis Model

In some parts of our subsequent analysis, we introduce the variable transformation (see Tao and Wang (2009); Tao and Winkler (2011, 2014b), Pang and Wang (2017))

$$\begin{aligned} a=ue^{-\xi w}, \end{aligned}$$

(3.3.1)

upon which (3.1.1) takes the form

$$\begin{aligned} \left\{ \begin{aligned} a_t=&e^{-\xi w}\nabla \cdot (e^{\xi w}\nabla a)-\chi e^{-\xi w}\nabla \cdot (e^{\xi w}a\nabla v)+\xi avw\\&+ a(\mu -\xi \eta w)(1-e^{\xi w}a-w),~&x\in \varOmega , t>0,\\ v_t=&\varDelta v +ae^{\xi w}- v,\quad&x\in \varOmega , t>0,\\ w_t=&- vw+\eta w(1-ae^{\xi w}-w) ,\quad&x\in \varOmega , t>0,\\ \frac{\partial a}{\partial \nu }=&\frac{\partial v}{\partial \nu }=0,\quad&x\in \partial \varOmega , t>0,\\ a(x,0):=&a_0(x)=u_0(x)e^{-\xi w_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad&x\in \varOmega .\\ \end{aligned}\right. \end{aligned}$$

(3.3.2)

The following lemma deals with local-in-time existence and the uniqueness of a classical solution for the problem (3.1.1).

Lemma 3.8

(Pang and Wang (2017)) Assume that the nonnegative functions $u_0,v_0,$ and $w_0$ satisfy (3.3.2) for some $\vartheta \in (0,1).$ Then there exists a maximal existence time $T_{max}\in (0,\infty ]$ and a triple of nonnegative functions

$$ \left\{ \begin{aligned}&\displaystyle {a\in C^0(\bar{\varOmega }\times [0,T_{max}))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max})),}\\&\displaystyle {v\in C^0(\bar{\varOmega }\times [0,T_{max}))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{max})),}\\&\displaystyle {w\in C^{2,1}(\bar{\varOmega }\times [0,T_{max})),}\\ \end{aligned}\right. $$

which solves (3.3.2) classically and satisfies

$$\begin{aligned} 0\le w\le \rho :=\max \{1,\Vert w_0\Vert _{L^\infty (\varOmega )}\} ~~\text{ in }~~ \varOmega \times (0,T_{max}). \end{aligned}$$

(3.3.3)

Moreover, if $T_{max}<+\infty$, then

$$\begin{aligned} \Vert a(\cdot , t)\Vert _{L^\infty (\varOmega )}+\Vert \nabla w(\cdot ,t)\Vert _{L^{5}(\varOmega )}\rightarrow \infty ~~ \text{ as }~~ t\nearrow T_{max}. \end{aligned}$$

(3.3.4)

In this subsection, we are going to establish an iteration step to develop the main ingredient of our result. Firstly, based on the ideas of Lemma 3.1 in Pang and Wang (2017) (see also Lemma 2.1 of Winkler (2010a)), we can derive the following properties of solutions of (3.1.1).

Lemma 3.9

Under the assumptions in Theorem 3.1, we derive that there exists a positive constant C such that the solution of (3.1.1) satisfies

$$ \int _{\varOmega }{u(x,t)}+\int _{\varOmega } {v^2}(x,t)+\int _{\varOmega }|\nabla {v}(x,t)|^2 \le C~~\text{ for } \text{ all }~~ t\in (0, T_{max}). $$

Lemma 3.10

Let

$$ {A}_1=\frac{1}{\delta +1}(\frac{\delta +1}{\delta })^{-\delta } [\frac{\delta (\delta -1)}{2}\chi ^2]^{\delta +1}C_7C_{\delta +1} $$

and $H(y)=y+ {A}_1y^{- \delta }$ for $y>0.$ For any fixed $\delta \ge 1,C_7,\chi ,C_{\delta +1}>0,$

$$\min _{y>0}H(y)=\frac{\delta (\delta -1)\chi ^2}{2}(C_7C_{\delta +1})^{\frac{1}{\delta +1}}.$$

Proof

It is easy to verify that $H'(y)=1- A_1\delta y^{-\delta -1}$ and $H'(\left( A_1\delta \right) ^{\frac{1}{\delta +1}})=0$. On the other hand, $\lim _{y\rightarrow 0^+}H(y)=+\infty$ and $\lim _{y\rightarrow +\infty }H(y)=+\infty$. Hence, we have

$$\begin{aligned} \min _{y>0}H(y)=H[\left( A_1\delta \right) ^{\frac{1}{\delta +1}}]=&\displaystyle {\frac{\delta (\delta -1)\chi ^2}{2}(C_7C_{\delta +1})^{\frac{1}{\delta +1}},}\\ \end{aligned} $$

whereby the proof is completed.

Lemma 3.11

Let $h(p) :=\displaystyle \frac{p\mu }{2}-\displaystyle \frac{p(p-1)\chi ^2}{2}(C_7C_{p+1})^{\frac{1}{p+1}}-(p-1)\xi \eta \rho ,$ where $p\ge 1,\xi ,\chi ,\eta ,\rho ,\mu ,C_7$ and $C_{p+1}$ are positive constants. Then there exists a positive constant $p_0>1$ such that

$$\begin{aligned} h(p_0)>0. \end{aligned}$$

(3.3.5)

Proof

Since $h(1)=\frac{\mu }{2}>0$, from the continuity of h it follows that for each $\mu >0$, there is some $p_0>1$ such that (3.3.5) holds.

According to the local existence results in Lemma 3.8, for any fixed $s\in (0,T_{max})$, it yields $(u(\cdot , s), v(\cdot , s), w(\cdot , s))\in (C^2(\bar{\varOmega }))^3$. Therefore, without loss of generality, we can assume that there exists a constant $\beta >0$ such that

$$\begin{aligned} \Vert u_0\Vert _{C^2(\bar{\varOmega })}\le \beta ,~~\Vert v_0\Vert _{C^2(\bar{\varOmega })}\le \beta ~~\text{ and }~~\Vert w_0\Vert _{C^2(\bar{\varOmega })}\le \beta . \end{aligned}$$

(3.3.6)

Lemma 3.12

Let $\mu ,\chi ,\eta$ and $\xi$ be the positive constants. Assume that (a, v, w) is a solution of (3.3.2) on $(0,T_{max})$. Then there exists a positive constant $C=C(p_0,|\varOmega |,\mu ,\chi ,\xi ,\eta ,\beta )$ such that

$$\begin{aligned} \int _{\varOmega }a^{p_0}(x,t)dx\le C ~~~\text{ for } \text{ all }~~ t\in (0,T_{max}), \end{aligned}$$

(3.3.7)

where $p_0>1$ is the same as in Lemma 3.11.

Proof

By using (3.3.2) and integration by parts, we get

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }e^{\xi w}a^{p_0}+(p_0+1)\int _{\varOmega }e^{\xi w}a^{p_0}\\ =\,&\displaystyle {\xi \int _{\varOmega }e^{\xi w}a^{p_0}\cdot \{-vw+\eta w(1-ae^{\xi w}-w)\}}\\&+\displaystyle {p_0\int _\varOmega e^{\xi w}a^{p_0-1}\cdot \{e^{-\xi w}\nabla \cdot (e^{\xi w}\nabla a)-\chi e^{-\xi w}\nabla \cdot (e^{\xi w}a\nabla v)\}}\\&+\displaystyle {a\xi vw +a(\mu -\xi \eta w)(1-ae^{\xi w}-w)\}+(p_0+1)\int _{\varOmega }e^{\xi w}a^{p_0}}\\ =\,&\displaystyle {-p_0(p_0-1)\int _{\varOmega } e^{\xi w}a^{p_0-2}|\nabla a|^2+p_0(p_0-1)\chi \int _{\varOmega } e^{\xi w}a^{p_0-1}\nabla a\cdot \nabla v}\\&+(p_0-1)\xi \int _\varOmega e^{\xi w}a^{p_0}vw\\&+\int _\varOmega e^{\xi w}a^{p_0}\{(p_0+1)+(p_0-1)\xi \eta w(w-1)+p_0\mu (1-w)\}\\&+\displaystyle {\int _\varOmega e^{2\xi w}a^{p_0+1}[(p_0-1)\xi \eta w-p_0\mu ]}\\ :\,=\,&\displaystyle {J_1+J_2+J_3+J_4+J_5~~\text{ for } \text{ all }~~ t\in (0,T_{max}).}\\ \end{aligned} \end{aligned}$$

(3.3.8)

Now, in light of (3.3.3), (3.3.4) and the Young inequality, we derive that

$$\begin{aligned} \displaystyle J_3\le&\displaystyle {\varepsilon _1\int _\varOmega e^{2\xi w}a^{p_0+1}+\frac{1}{p_0+1}\left( \varepsilon _1\cdot \frac{p_0+1}{p_0}\right) ^{-p_0} [(p_0-1)\xi ]^{p_0+1}\int _\varOmega e^{\xi w(1-p_0)}v^{p_0+1}} \nonumber \\ \le&\displaystyle {\varepsilon _1\int _\varOmega e^{2\xi w}a^{p_0+1}+\frac{1}{p_0+1}\left( \frac{\varepsilon _1(p_0+1)}{p_0}\right) ^{-p_0} [(p_0-1)\xi ]^{p_0+1}\int _\varOmega v^{p_0+1},} \end{aligned}$$

(3.3.9)

$$\begin{aligned} \displaystyle J_4\le&\displaystyle {[(p_0+1)+(p_0-1)\xi \eta \rho ^2+p_0\mu ]\int _\varOmega e^{\xi w}a^{p_0}} \nonumber \\ \le&\displaystyle {(p_0+1)[1+\xi \eta \rho ^2+\mu ]\int _\varOmega e^{\xi w}a^{p_0}} \\ \le&\displaystyle {\varepsilon _2\int _\varOmega e^{2\xi w}a^{p_0+1}} \displaystyle {+\frac{1}{p_0+1}\left( \frac{\varepsilon _2(p_0+1)}{p_0}\right) ^{-p_0} (p_0+1)^{p_0+1}[1+\xi \eta \rho ^2+\mu ]^{p_0+1}|\varOmega |} \nonumber \end{aligned}$$

(3.3.10)

as well as

$$ \displaystyle J_5\le \displaystyle {\int _\varOmega e^{2\xi w}a^{p_0+1}[(p_0-1)\xi \eta \rho -p_0\mu ]~~\text{ for } \text{ all }~~ t\in (0,T_{max})} $$

and

$$\begin{aligned} \displaystyle J_2\le&\displaystyle {\frac{p_0(p_0-1)}{2}\int _{\varOmega } e^{\xi w}a^{p_0-2}|\nabla a|^2+\frac{p_0(p_0-1)}{2}\chi ^2\int _{\varOmega } e^{\xi w}a^{p_0}|\nabla v|^2} \nonumber \\ \le&\displaystyle {\frac{p_0(p_0-1)}{2}\int _{\varOmega } e^{\xi w}a^{p_0-2}|\nabla a|^2+\lambda _0\int _\varOmega e^{2\xi w}a^{p_0+1}} \nonumber \\&\displaystyle {+\frac{1}{p_0+1}\left( \frac{\lambda _0(p_0+1)}{p_0}\right) ^{-p_0} \left[ \frac{p_0(p_0-1)}{2}\chi ^2\right] ^{p_0+1}\int _\varOmega e^{(1-p_0)\xi w}|\nabla v|^{2(p_0+1)} } \\ \le&\displaystyle {\frac{p_0(p_0-1)}{2}\int _{\varOmega } e^{\xi w}a^{p_0-2}|\nabla a|^2+\lambda _0\int _\varOmega e^{2\xi w}a^{p_0+1}} \nonumber \\&\displaystyle {+\frac{1}{p_0+1}\left( \frac{\lambda _0(p_0+1)}{p_0}\right) ^{-p_0} \left[ \frac{p_0(p_0-1)}{2}\chi ^2\right] ^{p_0+1}\int _\varOmega |\nabla v|^{2(p_0+1)} } \nonumber \end{aligned}$$

(3.3.11)

with any small positive constants $\varepsilon _1,\varepsilon _2$ and $\lambda _0.$

Inserting (3.3.9)–(3.3.11) into (3.3.8), we derive that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }e^{\xi w}a^{p_0}\\&+(p_0+1)\int _{\varOmega }e^{\xi w}a^{p_0}+\int _\varOmega e^{2\xi w}a^{p_0+1}[p_0\mu -\varepsilon _1-\varepsilon _2-\lambda _0-(p_0-1)\xi \eta \rho ]\\ \le&\displaystyle {\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}\int _\varOmega |\nabla v|^{2(p_0+1)}}\\&\displaystyle {+C_1(\varepsilon _1,\varepsilon _2)~~\text{ for } \text{ all }~~ t\in (0,T_{max}),}\\ \end{aligned} \end{aligned}$$

(3.3.12)

where

$$\begin{aligned} \begin{aligned} C_1(\varepsilon _1,\varepsilon _2):\,=\,&\displaystyle {\frac{1}{p_0+1}(\frac{\varepsilon _2(p_0+1)}{p_0})^{-p_0}(p_0+1)^{p_0+1} [1+\xi \eta \rho ^2+\mu ]^{p_0+1}|\varOmega |}\\&\displaystyle {+\frac{1}{p_0+1}(\frac{\varepsilon _1(p_0+1)}{p_0})^{-p_0} [(p_0-1)\xi ]^{p_0+1}\int _\varOmega v^{p_0+1}.}\\ \end{aligned} \end{aligned}$$

(3.3.13)

Next, from Lemma 3.9, $n=2$ and the Gagliardo–Nirenberg inequality, it follows that

$$\begin{aligned} \Vert v(\cdot ,t)\Vert _{L^{p_0}(\varOmega )}\le C_2~~\text{ for } \text{ all }~~~t\in (0,T_{max}). \end{aligned}$$

(3.3.14)

This along with (3.3.13) follows

$$ \begin{aligned} C_1(\varepsilon _1,\varepsilon _2)\le&C_3(\varepsilon _1,\varepsilon _2)\\ :\,=\,&\displaystyle {\frac{1}{p_0+1}(\varepsilon _2\times \frac{p_0+1}{p_0})^{-p_0} (p_0+1)^{p_0+1} [1+\xi \eta \rho ^2+\mu ]^{p_0+1}|\varOmega |}\\&\displaystyle {+C_2\frac{1}{p_0+1}(\varepsilon _1\times \frac{p_0+1}{p_0})^{-p_0} [(p_0-1)\xi ]^{p_0+1}.} \end{aligned} $$

From this and (3.3.12), we also obtain

$$ \begin{aligned}&\frac{d}{dt}\int _{\varOmega }e^{\xi w}a^{p_0}+(p_0+1)\int _{\varOmega }e^{\xi w}a^{p_0} \\&+\int _\varOmega e^{2\xi w}a^{p_0+1}[p_0\mu -\varepsilon _1-\varepsilon _2-\lambda _0-(p_0-1)\xi \eta \rho ] \\ \le&\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}\int _\varOmega |\nabla v|^{2(p_0+1)}\Vert \\&+C_3(\varepsilon _1,\varepsilon _2)~~\text{ for } \text{ all }~~ t\in (0,T_{max}). \end{aligned} $$

Then for any $t\in (0,T_{max})$, by means of the variation-of-constants representation for the above inequality, we can estimate

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }e^{\xi w}a^{p_0}(\cdot ,t)+[p_0\mu -\varepsilon _1-\varepsilon _2-\lambda _0-(p_0-1)\xi \eta \rho ] \\&\cdot \int _{0}^t\int _\varOmega e^{-(p_0-1)(t-s)}e^{2\xi w}a^{p_0+1} \\ \le&\int _{\varOmega }u^p_0+\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1} \\&\cdot \int _{0}^t\int _\varOmega e^{-(p_0-1)(t-s)}|\nabla v|^{2(p_0+1)} \\&+C_3(\varepsilon _1,\varepsilon _2)~~\text{ for } \text{ all }~~ t\in (0,T_{max}). \end{aligned} \end{aligned}$$

(3.3.15)

Next, according to the Gagliardo–Nirenberg inequality, (3.3.14) and Lemma 3.9, we can choose $C_4$ and $C_5$ such that

$$\begin{aligned} \begin{aligned} \displaystyle \Vert \nabla v(\cdot ,s)\Vert ^{2(p_0+1)}_{L^{2(p_0+1)}(\varOmega )}\le&\displaystyle { C_4\Vert v(\cdot ,s)\Vert ^{p_0+1}_{W^{2,p_0+1}(\varOmega )} \Vert \nabla v(\cdot ,s)\Vert ^{p_0+1}_{L^{2}(\varOmega )}}\\ \le&\displaystyle {C_5\Vert v(\cdot ,s)\Vert ^{p_0+1}_{W^{2,p_0+1}(\varOmega )} ~~\text{ for } \text{ all }~~ t\in (0,T_{max}).}\\ \end{aligned} \end{aligned}$$

(3.3.16)

Therefore, with the help of (3.3.16), applying (3.2.2) of Lemma 3.2 with $\gamma =p_0+1$, we obtain

$$\begin{aligned} \displaystyle&\displaystyle \displaystyle \frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}\int _{0}^t\int _\varOmega e^{-(p_0-1)(t-s)}|\nabla v|^{2(p_0+1)} \nonumber \\ \le&\displaystyle {\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}C_5\int _{0}^t e^{-(p_0-1)(t-s)}\Vert v(\cdot ,s)\Vert ^{p_0+1}_{W^{2,p_0+1}(\varOmega )}} \nonumber \\ \le&\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}C_5C_{p_0+1} \nonumber \\&\quad \cdot \int _{0}^t \int _\varOmega e^{-(p_0-1)(t-s)} u^{{{p_0}+1}}+C_6 \nonumber \\ \le&\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}C_5C_{p_0+1}e^{\xi (p_0-1)} \\&\quad \cdot \int _{0}^t \int _\varOmega e^{-(p_0-1)(t-s)} e^{2\xi w}a^{p_0+1}+C_6 \nonumber \\ \le&\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}C_7C_{p_0+1} \nonumber \\&\quad \cdot \int _{0}^t \int _\varOmega e^{-(p_0-1)(t-s)} e^{2\xi w}a^{p_0+1}+C_6 \nonumber \end{aligned}$$

(3.3.17)

for all $t\in (0,T_{max})$, where

$$ C_6:=\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}C_5C_{p_0+1}\Vert v_0\Vert ^{\gamma }_{W^{2,\gamma }(\varOmega )} $$

and

$$C_7:=C_5e^{\xi (p_0-1)}.$$

Substituting (3.3.17) into (3.3.15), we derive

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }e^{\xi w}a^{p_0}(\cdot ,t)+[p_0\mu -\varepsilon _1-\varepsilon _2-\lambda _0-(p_0-1)\xi \eta \rho ] \\&\cdot \int _{0}^t\int _\varOmega e^{-(p_0-1)(t-s)}e^{2\xi w}a^{p_0+1} \\ \le&\frac{1}{p_0+1}(\frac{\lambda _0(p_0+1)}{p_0})^{-p_0} [\frac{p_0(p_0-1)}{2}\chi ^2]^{p_0+1}C_7C_{p_0+1} \\&\cdot \int _{0}^t \int _\varOmega e^{-(p_0-1)(t-s)} e^{2\xi w}a^{p_0+1}+C_8(\varepsilon _1,\varepsilon _2)~~\text{ for } \text{ all }~~ t\in (0,T_{max}), \end{aligned} \end{aligned}$$

(3.3.18)

where $C_8(\varepsilon _1,\varepsilon _2):=C_3(\varepsilon _1,\varepsilon _2)+C_6.$ Choosing $\lambda _0=\left( A_1p_0\right) ^{\frac{1}{p_0+1}}$ in (3.3.18) and using Lemma 3.10, we derive

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }e^{\xi w}a^{p_0}(\cdot ,t)+[p_0\mu -\varepsilon _1-\varepsilon _2-\frac{p_0(p_0-1)\chi ^2}{2}(C_7C_{p_0+1})^{\frac{1}{p_0+1}}-(p_0-1)\xi \eta \rho ] \\&\cdot \int _{0}^t\int _\varOmega e^{-(p_0-1)(t-s)}e^{2\xi w}a^{p_0+1}\\ \le&\displaystyle {C_8(\varepsilon _1,\varepsilon _2)~~\text{ for } \text{ all }~~ t\in (0,T_{max}).}\\ \end{aligned} \end{aligned}$$

(3.3.19)

Now, for the above positive constants $\mu ,\chi ,\xi$ and $\eta$, due to Lemma 3.11, it has

$$\begin{aligned} p_0\mu -\frac{p_0(p_0-1)\chi ^2}{2}(C_7C_{p_0+1})^{\frac{1}{p_0+1}}-(p_0-1)\xi \eta \rho> \frac{p_0\mu }{2}>0, \end{aligned}$$

thus one can choose $\varepsilon _1$ and $\varepsilon _2$ appropriately small (e.g., $\varepsilon _1=\varepsilon _2=\frac{p_0\mu }{8}$) such that

$$\begin{aligned} 0<\varepsilon _1+\varepsilon _2<p_0\mu -\frac{p_0(p_0-1)\chi ^2}{2}(C_7C_{p_0+1})^{\frac{1}{p_0+1}}-(p_0-1)\xi \eta \rho . \end{aligned}$$

(3.3.20)

Collecting (3.3.19) and (3.3.20), we derive that there exists a positive constant $C_9$ such that

$$ \begin{aligned}&\displaystyle {\int _{\varOmega }u^{{p_0}}(x,t) dx\le C_9~~\text{ for } \text{ all }~~t\in (0, T_{max}).}\\ \end{aligned} $$

The proof of Lemma 3.12 is completed.

Lemma 3.13

Assume the hypothesis of Lemma 3.12 holds. Then for all $p>1$, there exists a positive constant $C=C(p,|\varOmega |,\mu ,\chi ,\xi ,\eta ,\beta )$ such that $\int _{\varOmega }a^p(x,t)dx\le C ~~~\text{ for } \text{ all }~~ t\in (0,T_{max}).$

Proof

Firstly, from Lemma 3.12 (see (3.3.7)) and (3.3.1), there exists a positive constant $C_1$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle {\int _{\varOmega }u^{{p_0}}(x,t) dx\le C_1~~\text{ for } \text{ all }~~t\in (0, T_{max}),}\\ \end{aligned} \end{aligned}$$

(3.3.21)

where $p_0>1$ is the same as that in Lemma 3.11. Next, we fix $q <\frac{2{p_0}}{(2-{p_0})^+}$ and choose some $\alpha > \frac{1}{2}$ such that

$$\begin{aligned} q <\frac{1}{\frac{1}{p_0}-\frac{1}{2}+\frac{2}{2}(\alpha -\frac{1}{2})}\le \frac{2{p_0}}{(2-{p_0})^+}. \end{aligned}$$

(3.3.22)

Now, involving the variation-of-constants formula for v, we have

$$\begin{aligned} v(t)=e^{-(A+1)}v_0 +\int _{0}^{t}e^{-(t-s)(A+1)}u(s) ds,~~ t\in (0, T_{max}). \end{aligned}$$

(3.3.23)

Hence, it follows from (3.3.6), (3.3.21)–(3.3.23) that

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert (A+1)^\alpha v(t)\Vert _{L^q(\varOmega )}}\\ \le&\displaystyle {c\int _{0}^{t}(t-s)^{-\alpha -\frac{2}{2}(\frac{1}{p_0}-\frac{1}{q})}e^{-\kappa (t-s)}\Vert u(s)\Vert _{L^{p_0}(\varOmega )}ds+ ce^{-\kappa t}t^{-\alpha +\frac{1}{q}}\Vert v_0\Vert _{L^{\infty }(\varOmega )}}\\ \le&\displaystyle {C_{2}\int _{0}^{+\infty }\sigma ^{-\alpha -\frac{2}{2}(\frac{1}{p_0}-\frac{1}{q})}e^{-\kappa \sigma }d\sigma +C_{3}t^{-\alpha +\frac{1}{q}}~~\text{ for } \text{ all }~~ t\in (0, T_{max}),}\\ \end{aligned} \end{aligned}$$

(3.3.24)

where $c>0$ is given by Lemma 3.1. Hence, in light of Lemmas 3.1 and 3.8, due to (3.3.22) and (3.3.24), we have

$$\begin{aligned} \int _{\varOmega }|\nabla {v}(t)|^{q}\le C_{4}~~\text{ for } \text{ all }~~ t\in (0, T_{max})~~\text{ and }~~q\in [1,\frac{2{p_0}}{(2-{p_0})^+}) \end{aligned}$$

(3.3.25)

with some positive constant $C_{4}.$ Now, due to the Sobolev embedding theorems and $N=2$, we conclude that

$$\begin{aligned} \Vert {v}(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C_{5}~~\text{ for } \text{ all }~~ t\in (0, T_{max}). \end{aligned}$$

(3.3.26)

Applying the Young inequality, one obtains from (3.3.3), (3.3.2) and (3.3.26) that for any $p>\max \{2,p_0-1\}$

$$\begin{aligned}&\displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }e^{\xi w}a^p+p(p-1)\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+p\mu \int _\varOmega e^{2\xi w}a^{p+1} \nonumber \\ =\,&\displaystyle {p(p-1)\chi \int _{\varOmega } e^{\xi w}a^{p-1}\nabla a\cdot \nabla v+(p-1)\xi \int _\varOmega e^{\xi w}a^{p}vw} \nonumber \\&+\displaystyle {\int _\varOmega e^{\xi w}a^{p}\{(p+1)+(p-1)\xi \eta w(w-1)+p\mu (1-w)\}} \nonumber \\&+\displaystyle {\int _\varOmega e^{2\xi w}a^{p+1}(p-1)\xi \eta w} \\ \le&\displaystyle {\frac{p(p-1)}{2}\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+\frac{p(p-1)}{2}\chi ^2\int _{\varOmega } e^{\xi w}a^{p}|\nabla v|^2+(p-1)\xi \int _\varOmega e^{\xi w}a^{p}vw} \nonumber \\&+\displaystyle {\int _\varOmega e^{\xi w}a^{p}\{(p+1)+(p-1)\xi \eta w(w-1)+p\mu (1-w)\}} \nonumber \\&+\displaystyle {\int _\varOmega e^{2\xi w}a^{p+1}(p-1)\xi \eta w} \nonumber \\ \le&\displaystyle {\frac{p(p-1)}{2}\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+\frac{p(p-1)}{2}\chi ^2\int _{\varOmega } e^{\xi w}a^{p}|\nabla v|^2+C_{6}\int _\varOmega a^{p+1}} \nonumber \\ \le&\frac{p(p-1)}{2}\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+\frac{p(p-1)}{2}\chi ^2e^{\xi \rho }\int _{\varOmega } a^{p}|\nabla v|^2 \nonumber \\&+C_{6}\int _\varOmega a^{p+1}~~\text{ for } \text{ all }~~ t\in (0,T_{max}). \nonumber \end{aligned}$$

(3.3.27)

Next, with the help of the Gagliardo–Nirenberg inequality (see, e.g., Zheng (2015)), it yields that

$$ \begin{aligned} C_{6}\displaystyle \int _\varOmega a^{p+1}=\,&\displaystyle { C_{6}\Vert {{a^{\frac{p}{2}}}}\Vert ^{2\frac{(p+1)}{p}}_{L^{2\frac{(p+1)}{p} }(\varOmega )}}\\ \le&\displaystyle {C_{7}(\Vert \nabla {{a^{\frac{p}{2}}}}\Vert _{L^2(\varOmega )}^{\mu _1}\Vert {{a^{\frac{p}{2}}}}\Vert _{L^\frac{2p_0}{p}(\varOmega )}^{1-\mu _1}+\Vert {{a^{\frac{p}{2}}}}\Vert _{L^\frac{2p_0}{p}(\varOmega )})^{2\frac{(p+1)}{p}}}\\ \le&\displaystyle {C_{8}(\Vert \nabla {{a^{\frac{p}{2}}}}\Vert _{L^2(\varOmega )}^{2\mu _1}+1)}\\ =\,&\displaystyle {C_{8}(\Vert \nabla {{a^{\frac{p}{2}}}}\Vert _{L^2(\varOmega )}^{\frac{2(p-p_0+1)}{p+1}}+1)}\\ \end{aligned} $$

with some positive constants $C_{7}, C_{8}$ and

$$\mu _1=\frac{\frac{{p}}{p_0}-\frac{p}{p+1}}{\frac{{p}}{p_0}}= \frac{p+1-p_0}{p+1}\in (0,1).$$

Since, $p_0>1$ yields $p_0<\frac{2{p_0}}{2(2-{p_0})^+}$, in light of the Hölder inequality and (3.3.25), we derive

$$ \begin{aligned} \displaystyle \frac{\chi ^2p({p}-1)}{2}e^{\xi \rho }\displaystyle \int _\varOmega {{a^{p }}} |\nabla {v}|^2\le&\displaystyle { \displaystyle \frac{\chi ^2p({p}-1)}{2}e^{\xi \rho }\left( \displaystyle \int _\varOmega {{a^{\frac{p_0}{p_0-1} p }}}\right) ^{\frac{p_0-1}{p_0}}\left( \displaystyle \int _\varOmega |\nabla {v}|^{2p_0}\right) ^{\frac{1}{p_0}}}\\ \le&\displaystyle {C_{9}\Vert {{a^{\frac{p}{2}}}}\Vert ^{2}_{L^{2\frac{p_0}{p_0-1} }(\varOmega )},}\\ \end{aligned} $$

where $C_{9}$ is a positive constant. Since $p_0> 1$ and $p>p_0-1$, we have

$$\frac{p_0}{p}\le \frac{p_0}{p_0-1}<+\infty ,$$

which together with the Gagliardo–Nirenberg inequality (see, e.g., Zheng (2015)) implies that

$$ \begin{aligned} C_{9}\Vert {{a^{\frac{p}{2}}}}\Vert ^{2}_{L^{2\frac{p_0}{p_0-1} }(\varOmega )}\le&\displaystyle {C_{10}(\Vert \nabla {{a^{\frac{p}{2}}}}\Vert _{L^2(\varOmega )}^{\mu _2}\Vert {{a^{\frac{p}{2}}}}\Vert _{L^\frac{2p_0}{p}(\varOmega )}^{1-\mu _2}+\Vert {{a^{\frac{p}{2}}}}\Vert _{L^\frac{2p_0}{p}(\varOmega )})^{2}}\\ \le&\displaystyle {C_{11}(\Vert \nabla {{a^{\frac{p}{2}}}}\Vert _{L^2(\varOmega )}^{2\mu _2}+1)}\\ =\,&\displaystyle {C_{11}(\Vert \nabla {{a^{\frac{p}{2}}}}\Vert _{L^2(\varOmega )}^{\frac{2(p-p_0+1)}{p}}+1)}\\ \end{aligned} $$

with some positive constants $C_{10}, C_{11}$ and

$$\mu _2=\frac{\frac{{p}}{p_0}-\frac{p}{\frac{p_0}{p_0-1} p }}{\frac{{p}}{p_0}}\in (0,1).$$

Moreover, an application of the Young inequality shows that

$$\begin{aligned} \begin{aligned}&C_{6}\displaystyle \int _\varOmega a^{p+1}+\displaystyle \frac{\chi ^2p({p}-1)}{2}e^{\xi \rho }\displaystyle \int _\varOmega a^{{p}}|\nabla v|^2 \\ \le&\displaystyle { \frac{p({{p}-1})}{4}\int _{\varOmega }a^{p-2}|\nabla a|^2+C_{12}} \\ \le&\displaystyle { \frac{p({{p}-1})}{4}\int _{\varOmega }e^{\xi w}a^{{{p}-2}}|\nabla a|^2+C_{12}.}\\ \end{aligned} \end{aligned}$$

(3.3.28)

Inserting (3.3.28) into (3.3.27), we conclude that

$$ \begin{aligned}&\displaystyle {\frac{d}{dt}\displaystyle \int _{\varOmega }e^{\xi w}a^p+\frac{p({{p}-1})}{4}\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+p\mu \int _\varOmega e^{2\xi w}a^{p+1}\le C_{13}.}\\ \end{aligned} $$

Therefore, integrating the above inequality with respect to t yields

$$ \begin{aligned} \Vert a(\cdot , t)\Vert _{L^{{p}}(\varOmega )}\le C_{14} ~~ \text{ for } \text{ all }~~p\ge 1~~\text{ and }~~ t\in (0,T_{max}) \\ \end{aligned} $$

for some positive constant $C_{14}$.

Remark 3.2

It only assumes that $\mu >0$ which is different from that in Pang and Wang (2017). Indeed, by the technical lemma (see Lemma 3.10), one could conclude the boundedness of $\int _{\varOmega }{a^{q_0}}~(\text{ for } \text{ some }~q_0 > 1),$, and further in light of the variation-of-constants formula and $L^q$-$L^p$ estimates for the heat semigroup, one may derive the boundedness of $\int _{\varOmega }{a^{p}}$ (for any $p> 1$).

Our main result on global existence and boundedness thereby becomes a straightforward consequence of Lemma 3.8 and Lemma 3.13.

The proof of Theorem 3.1: The proof of Theorem 3.1 consists of the following steps.

Step 1. $\Vert a(\cdot , t)\Vert _{L^{\infty }(\varOmega )}$: Firstly, in light of (3.3.3), due to Lemma 3.13, we derive that there exist positive constants $p_0>2$ and $C_1$ such that

$$\begin{aligned} \begin{aligned} \Vert u(\cdot , t)\Vert _{L^{{p_0}}(\varOmega )}\le C_{1} ~~ \text{ for } \text{ all }~~t\in (0,T_{max}). \\ \end{aligned} \end{aligned}$$

Next, since $p_0>2$ and $n=2$ yield to $+\infty =\frac{np_0}{(n-p_0)_{+}}$, therefore, by using Lemma 3.1 (see also Lemma 2.1 of Ishida et al. (2014)), we conclude that

$$\begin{aligned} \begin{aligned}&\displaystyle {\Vert \nabla v(t)\Vert _{L^\infty (\varOmega )}\le C_2~~\text{ for } \text{ all }~~ t\in (0, T_{max}).}\\ \end{aligned} \end{aligned}$$

(3.3.29)

Applying the Young inequality, in light of (3.3.3) and the first equation of (3.3.2), one obtains from (3.3.29) that for any $p\ge 4$

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }e^{\xi w}a^p+p(p-1)\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+\int _{\varOmega }e^{\xi w}a^p\\ =\,&\displaystyle {\xi \int _{\varOmega }e^{\xi w}a^p\cdot \{-vw+\eta w(1-ae^{\xi w}-w)\}}\\&+\displaystyle {p\int _\varOmega e^{\xi w}a^{p-1}\cdot \{e^{-\xi w}\nabla \cdot (e^{\xi w}\nabla a)-\chi e^{-\xi w}\nabla \cdot (e^{\xi w}a\nabla v)\}}\\&+\displaystyle {a\xi vw +a(\mu -\xi \eta w)(1-ae^{\xi w}-w)\}+p\int _{\varOmega }e^{\xi w}a^p}\\ \le&\displaystyle {\frac{p(p-1)}{4}\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+p(p-1)\chi ^2C_3\int _{\varOmega } e^{\xi w}a^{p}}\\&+\displaystyle {(p-1)\xi \int _\varOmega e^{\xi w}a^{p}vw+\int _\varOmega e^{\xi w}a^{p}\{(p+1)+(p-1)\xi \eta w(w-1)+p\mu (1-w)\}}\\&+\displaystyle {\int _\varOmega e^{2\xi w}a^{p+1}[(p-1)\xi \eta w-p\mu ]}\\ \le&\displaystyle {\frac{p(p-1)}{4}\int _{\varOmega } e^{\xi w}a^{p-2}|\nabla a|^2+C_4p^2(\int _{\varOmega } a^{p+1}+1)~~\text{ for } \text{ all }~~ t\in (0,T_{max}),}\\ \end{aligned} \end{aligned}$$

(3.3.30)

where $C_3>0$ and $C_4>0$ are independent of p. Here and throughout the proof of Theorem 3.1, we shall denote by $C_i(i\in \mathbb {N})$ the several positive constants independent of p. Therefore, (3.3.30) implies that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\displaystyle {\int _{\varOmega }e^{\xi w}a^p+C_5\int _{\varOmega } |\nabla a^{\frac{p}{2}}|^2+\int _{\varOmega }e^{\xi w}a^p\le C_4p^2(\int _{\varOmega } a^{p+1}+1)~~\text{ for } \text{ all }~~ t\in (0,T_{max}).}\\ \end{aligned} \end{aligned}$$

(3.3.31)

Next, once more by means of the Gagliardo–Nirenberg inequality, we can estimate

$$\begin{aligned} \begin{aligned} C_4p^2 \displaystyle \int _\varOmega a^{p+1}=\,&\displaystyle { C_4p^2\Vert a^{\frac{p}{2}}\Vert _{L^\frac{2(p+1)}{p}(\varOmega )}^{\frac{2(p+1)}{p}} }\\ \le&\displaystyle { C_6p^2(\Vert \nabla a^{\frac{p}{2}}\Vert _{L^{2}(\varOmega )}^{\frac{2(p+1)}{p}\varsigma _1} \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2(p+1)}{p}(1-\varsigma _1)}+ \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2(p+1)}{p}}) }\\ =\,&\displaystyle { C_6p^2(\Vert \nabla a^{\frac{p}{2}}\Vert _{L^{2}(\varOmega )}^{\frac{p+2}{p}} \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}+ \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2(p+1)}{p}})}\\ \le&\displaystyle { C_5\Vert \nabla a^{\frac{p}{2}}\Vert _{L^{2}(\varOmega )}^{2}+C_7p^{\frac{4p}{p-2}} \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2p}{p-2}}+ C_6p^2\Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2(p+1)}{p}}}\\ \le&\displaystyle { C_5\Vert \nabla a^{\frac{p}{2}}\Vert _{L^{2}(\varOmega )}^{2}+C_8p^{\frac{4p}{p-2}} \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2p}{p-2}},}\\ \end{aligned} \end{aligned}$$

(3.3.32)

where

$$0<\varsigma _1=\frac{2-\frac{2p}{2(p+1)}}{1-\frac{2}{2}+2}=\frac{p+2}{2(p+1)}<1.$$

Here, we have used the fact that $\frac{4p}{p-2}\ge 2.$ Therefore, inserting (3.3.32) into (3.3.31), we derive that

$$\begin{aligned} \begin{aligned} \displaystyle \frac{d}{dt}\displaystyle \int _{\varOmega }e^{\xi w}a^p+\int _{\varOmega }e^{\xi w}a^p\le&\displaystyle { C_8p^{\frac{4p}{p-2}} \Vert a^{\frac{p}{2}}\Vert _{L^1(\varOmega )}^{\frac{2p}{p-2}}+C_4p^2}\\ \le&\displaystyle { C_9p^{\frac{4p}{p-2}}\left( \max \{1, \Vert u^{\frac{p}{2}}\Vert _{L^1(\varOmega )}\right) ^{\frac{2p}{p-2}}.}\\ \end{aligned} \end{aligned}$$

(3.3.33)

Now, choosing $p_i=2^{i+2}$ and letting $M_i =\max \{1,\sup _{t\in (0,T)}\int _{\varOmega } a^{\frac{{p_i}}{2}}\}$ for $T\in (0, T_{max})$ and $i= 1, 2,3,\ldots$, we then obtain from (3.3.33) that

$$ \begin{aligned}&\displaystyle {\frac{d}{dt}\int _{\varOmega }e^{\xi w}a^{p_i}+\int _{\varOmega }e^{\xi w}a^{p_i}\le C_{10}{p_i^{\frac{2p_i}{p_i-2}}} M^{\frac{2p_i}{p_i-2}}_{i-1}(T),}\\ \end{aligned} $$

which together with the comparison argument entails that there exists $\lambda >1$ independent of i such that

$$\begin{aligned} \begin{aligned}&\displaystyle {M_{i}(T)\le \max \{\lambda ^iM^{\frac{2p_i}{p_i-2}}_{i-1}(T),e^{\xi }|\varOmega |\Vert a_0\Vert _{L^\infty (\varOmega )}^{p_i}\}.}\\ \end{aligned} \end{aligned}$$

(3.3.34)

Here, we use the fact that $\kappa _i:=\frac{2p_i}{p_i-2}\le 4.$ Now, if $\lambda ^iM^{\kappa _i}_{i-1}(T)\le e^{\xi \rho }|\varOmega |\Vert a_0\Vert _{L^\infty (\varOmega )}^{p_i}$ for infinitely many $i\ge 1$, we get

$$\left( \sup _{t\in (0,T)}\int _{\varOmega } a^{p_{i-1}}(\cdot ,t)\right) ^{\frac{1}{p_{i-1}}}\le \left( \frac{e^{\xi \rho }|\varOmega |\Vert a_0\Vert _{L^\infty (\varOmega )}^{p_i}}{\lambda ^i}\right) ^{\frac{1}{p_{i-1}\kappa _i}} $$

for such i, which entails that

$$\begin{aligned} \sup _{t\in (0,T)}\Vert a(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \Vert a_0\Vert _{L^\infty (\varOmega )}. \end{aligned}$$

(3.3.35)

Otherwise, if $\lambda ^iM^{\kappa _i}_{i-1}(T)>e^{\xi }|\varOmega |\Vert a_0\Vert _{L^\infty (\varOmega )}^{p_i}$ for all sufficiently large i, then by (3.3.34), we derive that

$$\begin{aligned} \displaystyle {M_{i}(T)\le \lambda ^iM^{\kappa _i}_{i-1}(T)~~~\text{ for } \text{ all } \text{ sufficiently } \text{ large }~i,} \end{aligned}$$

(3.3.36)

and thus (3.3.36) is still valid for all $i\ge 1$ upon enlarging $\lambda$ if necessary. That is,

$$ \begin{aligned}&\displaystyle {M_{i}(T)\le \lambda ^iM^{\kappa _i}_{i-1}(T)~~~\text{ for } \text{ all }~i\ge 1.}\\ \end{aligned} $$

Therefore, based on a straightforward induction (see, e.g., Lemma 3.12 of Tao and Winkler (2014b)), we have

$$\begin{aligned} \begin{aligned} \displaystyle M_{i}(T)\le&\displaystyle { \lambda ^{i+\sum _{j=2}^i(j-1)\cdot \varPi _{k=j}^i\kappa _k}M_{0}^{\varPi _{k=1}^i\kappa _k}~~~\text{ for } \text{ all }~ i \ge 1,}\\ \end{aligned} \end{aligned}$$

(3.3.37)

where $\kappa _k = 2(1+\varepsilon _k)$ satisfies $\varepsilon _k =\frac{4}{p_k-2}\le \frac{C_{11}}{2^k}$ for all $k\ge 1$ with some $C_{11}> 0$. Therefore, due to the fact that $\ln (1 + x) \le x (x\ge 0)$, we derive

$$ \begin{aligned} \varPi _{k=j}^i\kappa _k=\,&\displaystyle {2^{i+1-j}e^{\Sigma _{k=j}^i\ln (1+\varepsilon _j)}}\\ \le&\displaystyle {2^{i+1-j}e^{\Sigma _{k=j}^i\varepsilon _j}}\\ \le&\displaystyle {2^{i+1-j}e^{C_{11}}~~~\text{ for } \text{ all }~~ i \ge 1~~~\text{ and }~~ j\in \{1,\ldots ,i\},}\\ \end{aligned} $$

which implies that

$$ \begin{aligned} \displaystyle \frac{\sum _{j=2}^i(j-1)\cdot \varPi _{k=j}^i\kappa _k}{2^{i+2}}\le&\displaystyle {\frac{\sum _{j=2}^i(j-1)2^{i+1-j}e^{C_{11}}}{2^{i+2}}}\\ \le&\displaystyle {\frac{e^{C_{11}}}{2}\sum _{j=2}^i\frac{(j-1)}{2^{j}}}\\ \le&\displaystyle {\frac{3e^{C_{11}}}{8}.} \end{aligned} $$

By the definition of $p_i$, we easily deduce from (3.3.37) that

$$ \begin{aligned} \displaystyle M_{i}^{\frac{1}{p_i}}(T)\le&\displaystyle { \lambda ^{\frac{i}{2^{i+2}}+\frac{\sum _{j=2}^i(j-1)\cdot \varPi _{k=j}^i\kappa _k}{2^{i+2}}}M_{0}^{\frac{\varPi _{k=1}^i\kappa _k}{2^{i+2}}}} \le \displaystyle { \lambda ^{\frac{i}{2^{i+2}}}\lambda ^{\frac{3e^{C_{11}}}{8}} M_{0}^{\frac{e^{C_{11}}}{4}},}\\ \end{aligned} $$

which after taking $i\rightarrow \infty$ and $T\nearrow T_{max}$ readily implies that

$$\begin{aligned} \Vert a(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le \lambda ^{\frac{3e^{C_{11}}}{8}} M_{0}^{\frac{e^{C_{11}}}{4}}~~~\text{ for } \text{ all }~t\in (0,T_{max}). \end{aligned}$$

(3.3.38)

Step 2: $\Vert \nabla w(\cdot ,t)\Vert _{L^5(\varOmega )}$ Employing almost exactly the same arguments as that in the proof of Lemmas 3.5–3.6 in Pang and Wang (2017) (the minor necessary changes are left as an easy exercise to the reader), and taking advantage of (3.3.29) and (3.3.38), we conclude the estimate for any $T<T_{max},$

$$\Vert \nabla w(\cdot ,t)\Vert _{L^5(\varOmega )}\le C~~~\text{ for } \text{ all }~t\in (0,T). $$

Now, with the above estimate in hand, using (3.3.35) and (3.3.38), employing the extendibility criterion provided by Lemma 3.8, we may prove Theorem 3.1.

Remark 3.3

If $\mu >\xi \eta \max \{\Vert u_0\Vert _{L^\infty (\varOmega )},1\}+\mu ^*(\chi ^2,\xi )$ (see the proof of Lemma 3.4 to Pang and Wang (2017)), one only needs to estimate $Cp^2\int _{\varOmega } a^{p}$ other than $Cp^2(\int _{\varOmega } a^{p+1}+1).$

3.4 Global Boundedness of Solutions to a Chemotaxis–Haptotaxis Model with Tissue Remodeling

3.4.1 A Convenient Extensibility Criterion

For the convenience in some parts of our subsequent analysis, we introduce the variable transformation Tao and Wang (2009, 2008); Tao and Winkler (2014b)

$$a = ue^{-\xi w}, $$

upon which (3.1.1) takes the following form:

$$\begin{aligned} \left\{ \begin{aligned}&a_t=e^{-\xi w}\nabla \cdot (e^{\xi w}\nabla a)-e^{-\xi w}\chi \nabla \cdot ( e^{\xi w}a \nabla v)+\xi a v w&\\&\quad \quad +a \mu (r-e^{\xi w}a-w)-a\xi \eta w(1-e^{\xi w}a-w),&x\in \varOmega ,t>0, \\&\sigma v_t=\varDelta v-v+ e^{\xi w}a,&x\in \varOmega ,t>0, \\&w_t=-vw +\eta w(1-w-e^{\xi w}a),&x\in \varOmega ,t>0, \\&\displaystyle \frac{ \partial a}{\partial \nu }=\frac{ \partial v}{\partial \nu }=\frac{ \partial w}{\partial \nu }=0,&x\in \partial \varOmega ,t>0, \\&a(x,0)=a_0(x)=u_0(x) e^{-\xi w_0(x)}, \sigma v_0(x,0)= \sigma v_0(x),&w(x,0)=w_0(x),\quad x\in \varOmega . \end{aligned} \right. \end{aligned}$$

(3.4.1)

We note that (3.1.1) and (3.4.1) are equivalent within the concept of classical solutions.

The following result is concerned with the local existence and uniqueness of classical solutions to the problem (3.4.1), along with a convenient extensibility criterion for such solutions.

Lemma 3.14

Let $\chi>0, \xi >0$, $\mu >0$ and $r>0$, and suppose that $u_0, v_0$ and $w_0$ satisfy (3.1.2) with some $\vartheta \in (0,1).$ Then the problem (3.4.1) admits a unique classical solution

$$\begin{aligned} \left\{ \begin{aligned} a\in C^0(\bar{\varOmega }\times [0,T_{max})) \cap C^{2,1}(\bar{\varOmega }\times (0,T_{max})) \\ v\in C^0(\bar{\varOmega }\times [0,T_{max})) \cap C^{2,1}(\bar{\varOmega }\times (0,T_{max}))\\ w\in C^{2,1}(\bar{\varOmega }\times [0,T_{max})) \end{aligned}\right. \end{aligned}$$

(3.4.2)

with $a\ge 0, v\ge 0$ and $0\le w\le A:= \max \{1, \Vert w_0\Vert _{L^\infty (\varOmega )}\}$, where $T_{max}$ denotes the maximal existence time. In addition, if $T_{max}<+\infty$, then

$$\begin{aligned} \Vert a(\cdot , t)\Vert _{L^\infty (\varOmega )}+\Vert \nabla w(\cdot ,t)\Vert _{L^5(\varOmega )} \rightarrow \infty ~~ \text{ as }~~ t\nearrow T_{max}. \end{aligned}$$

(3.4.3)

Proof

Invoking well-established fixed point arguments and applying the standard parabolic regularity theory, one can readily verify the local existence and uniqueness of classical solutions, as well as the extensibility criterion (3.4.3) (see Pang and Wang (2017); Tao and Winkler (2014a, 2014b) for instance). With the help of the maximum principle, we can also verify the asserted nonnegativity of the solutions.

It should be pointed out that the extensibility criterion in (3.4.3) involves the $L^5$-norm of $|\nabla w|$. Although the $L^5$-norm of $|\nabla w|$ is time-dependent, it is sufficient to enable us to apply standard parabolic regularity theory to the first equation of (3.4.1) in the two-dimensional setting (see Lemma 2.2 of Pang and Wang (2017) and Tao and Winkler (2014b) for instance).

For the classical solution of (3.4.1), the following observation will be used frequently below.

Lemma 3.15

Let (a, v, w) be the classical solution of (3.4.1) in $\varOmega \times [0,T_{max})$. Then for any $p>1$, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^p+\displaystyle \frac{2(p-1)}{p} \int _\varOmega e^{\xi w}|\nabla a^{\frac{p}{2} }|^2+(p \mu -(p-1)\xi \eta A) \displaystyle \int _\varOmega a^{p+1}e^{2\xi w}\\ \le&\displaystyle \frac{\chi ^2 p(p-1)}{2} \int _\varOmega e^{\xi w}a^p |\nabla v|^2+\xi A(p-1)\int _\varOmega e^{\xi w}a^p v \\&+(\mu pr+ \xi \eta A^2(p-1))\int _\varOmega e^{\xi w}a^p \end{aligned} \end{aligned}$$

(3.4.4)

with $A= \max \{1, \Vert w_0\Vert _{L^\infty (\varOmega )}\}$.

Proof

Testing the first equation in (3.4.1) by $a^{p-1}$ with $p>1$ and integrating by parts yields

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^p+\displaystyle \frac{4(p-1)}{p} \int _\varOmega e^{\xi w}|\nabla a^{\frac{p}{2} }|^2+(p \mu -(p-1)\xi \eta A) \displaystyle \int _\varOmega a^{p+1}e^{2\xi w}\\ \le&\displaystyle \chi p(p-1) \int _\varOmega e^{\xi w}a^{p-1} \nabla a \cdot \nabla v+\xi A(p-1)\int _\varOmega e^{\xi w}a^p v \\&+(\mu pr+ \xi \eta A^2(p-1))\int _\varOmega e^{\xi w}a^p. \end{aligned} \end{aligned}$$

(3.4.5)

Here, we note that $0\le w \le A$ in $\varOmega \times [0,T_{max})$. By the Young inequality, we estimate

$$ \begin{aligned}&\displaystyle \chi p(p-1) \int _\varOmega e^{\xi w}a^{p-1} \nabla a \cdot \nabla v\\ \le&\displaystyle \frac{p(p-1)}{2} \int _\varOmega e^{\xi w} a^{p-2}|\nabla a|^2+ \displaystyle \frac{\chi ^2 p(p-1)}{2} \int _\varOmega e^{\xi w}a^p |\nabla v|^2. \end{aligned} $$

This together with (3.4.5) proves (3.4.4).

3.4.2 Global Existence in Two-Dimensional Domains

According to Lemma 2.6, the key step in the proof of Theorem 3.2 is to establish a priori estimates of $\Vert a(\cdot , t)\Vert _{L^\infty (\varOmega )}$ and $\Vert \nabla w(\cdot ,t)\Vert _{L^5(\varOmega )}$. As pointed out in Tao and Winkler (2014b), one essential analytic difficulty stems from the fact that the chemotaxis and haptotaxis terms in the first equation in (3.1.1) require different $L^p$-estimate techniques, since ECM density satisfies an ordinary differential equation (ODE) whereas MDE concentration satisfies a parabolic equation (PDE). This part establishes the crucial a priori estimates of solutions via identifying a certain dissipative property of the functionals $\int _\varOmega e^{\xi w}a^2$ and $\int _\varOmega e^{\xi w}a\ln a$ with $a=e^{-\xi w}u$.

1. The Case of $\sigma =1$

According to the above local existence result, $(u(\cdot ,s),v(\cdot ,s), w(\cdot ,s))\in (C^2(\bar{\varOmega }))^3$ for any $s\in (0,T_{max})$. Hence without loss of generality, we may assume that there exists a constant $C>0$ such that

$$\begin{aligned} \Vert u_0\Vert _{C^2(\bar{\varOmega })}+\Vert v_0\Vert _{C^2(\bar{\varOmega })}+\Vert w_0\Vert _{C^2(\bar{\varOmega })}\le C. \end{aligned}$$

(3.4.6)

From now on, (u, v, w) is the unique maximal solution provided by Lemma 3.14. In order to avoid regularity problems, we assume in the rest of this section that the initial data satisfies (3.1.2). Some basic but important properties of solutions of (3.1.1) are summarized in the next lemmas.

Lemma 3.16

Let (u, v, w) be the classical solution of (3.1.1) with $\sigma =1$. Then we have

(i) $\Vert u(\cdot ,t)\Vert _{L^1(\varOmega )}\le m_0:=\max \{r|\varOmega |, \Vert u_0\Vert _{L^1(\varOmega )}\}~~~~\text{ for } \text{ all }~~ t\in (0, T_{max});$

(ii) $\displaystyle \int ^{t+\tau }_t \Vert u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le m_1:=r^2|\varOmega |+\frac{2m_0}{\mu }$ for any $0<\tau \le \min \{1,\frac{ T_{max}}{3}\}$ and all $t\in (0,T_{max}-\tau )$;

(iii) $\Vert v(\cdot ,t)\Vert _{L^1(\varOmega )}\le \tilde{m}_0:=\max \{m_0, \Vert v_0\Vert _{L^1(\varOmega )}\} ~~~~\text{ for } \text{ all }~~ t\in (0, T_{max});$

(iv) $\Vert \nabla v(\cdot , t)\Vert ^2_{L^2(\varOmega )}\le m_2:= \displaystyle \frac{r\mu +2}{\mu }m_0+ \Vert \nabla v_0\Vert ^2_{L^2(\varOmega )} ~~~\text{ for } \text{ all }~~ t\in (0, T_{max});$

(v) $\displaystyle \int ^{t+\tau }_t \Vert \varDelta v(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le m_3:=m_2+ m_1$ for any $0<\tau \le \min \{1,\frac{ T_{max}}{3}\}$ and all $t\in (0,T_{max}-\tau )$.

Proof

(i) Integrating the first equation in (3.1.1) with respect to $x\in \varOmega$ yields

$$\begin{aligned} \displaystyle {\frac{d}{dt}} \int _{\varOmega }u(x,t) \le r\mu \int _{\varOmega }u(x,t)-\mu \int _{\varOmega }u^2(x,t), \end{aligned}$$

(3.4.7)

since $w\ge 0$ by Lemma 3.14. Moreover, by $2 \mu r u \le \mu u^2 + \mu r^2$, we get

$$ \displaystyle {\frac{d}{dt}} \int _{\varOmega }u(x,t)+r \mu \int _{\varOmega }u(x,t)\le \mu r^2 |\varOmega |, $$

which implies that $\Vert u(\cdot ,t)\Vert _{L^1(\varOmega )}\le \max \{r|\varOmega |, \Vert u_0\Vert _{L^1(\varOmega )}\}$.

(ii) By (3.4.7) and the Cauchy–Schwartz inequality, we also have

$$\begin{aligned} \displaystyle {\frac{d}{dt}} \int _{\varOmega }u(x,t)+\frac{\mu }{2} \int _{\varOmega }u^2(x,t)\le \frac{\mu r^2}{2} |\varOmega |. \end{aligned}$$

(3.4.8)

Then we integrate (3.4.8) over $(t,t+\tau )$ to get

$$ \frac{\mu }{2} \displaystyle \int ^{t+\tau }_t \int _{\varOmega }u^2\le \frac{r^2\mu }{2} |\varOmega |\tau +\int _{\varOmega }u(x,t), $$

which along with (i) yields (ii) of the lemma.

(iii) Integrating the second equation in (3.1.1) with respect to $x\in \varOmega$ yields

$$ \displaystyle {\frac{d}{dt}} \int _{\varOmega }v(x,t)+ \int _{\varOmega }v(x,t) \le \int _{\varOmega }u(x,t)\le \sup _{t\ge 0} \int _{\varOmega }u(x,t). $$

So (iii) follows from the nonnegativity of v and (i).

(iv) Multiplying the second equation in (3.1.1) by $-\varDelta v$ and integrating over $\varOmega$, we find

$$ \begin{aligned}&\displaystyle \frac{1}{2} \displaystyle {\frac{d}{dt}} \int _{\varOmega }|\nabla v(x,t)|^2+ \int _{\varOmega }| \varDelta v(x,t)|^2+ \int _{\varOmega }|\nabla v(x,t)|^2\\ =\,&-\displaystyle \int _{\varOmega }u \triangle v\\ \le&\displaystyle \frac{1}{2} \int _{\varOmega }| \varDelta v(x,t)|^2+\frac{1}{2} \int _{\varOmega } u^2(x,t) \end{aligned} $$

and thus

$$\begin{aligned} \displaystyle {\frac{d}{dt}} \int _{\varOmega }|\nabla v(x,t)|^2+ \displaystyle \int _{\varOmega }| \varDelta v(x,t)|^2 +\int _{\varOmega }|\nabla v(x,t)|^2\le \int _{\varOmega } u^2(x,t). \end{aligned}$$

(3.4.9)

Combining (3.4.9) with (3.4.7), we can obtain

$$\begin{aligned} \begin{aligned}&\displaystyle {\frac{d}{dt}}\int _{\varOmega } (u(x,t)+\mu |\nabla v(x,t)|^2) + \displaystyle \int _{\varOmega } (u(x,t)+ \mu |\nabla v(x,t)|^2) \\ \le&(r\mu +1)\displaystyle \int _{\varOmega }u(x,t)\\ \le&(r\mu +1)m_0, \end{aligned} \end{aligned}$$

(3.4.10)

which, together with the Gronwall lemma, yields

$$ \mu \int _{\varOmega }|\nabla v(x,t)|^2\le \displaystyle (r\mu +2)m_0+ \mu \Vert \nabla v_0\Vert ^2_{L^2(\varOmega )} $$

and hence (iv) holds.

(v) In view of (3.4.9), we have

$$ \int ^{t+\tau }_{t}\displaystyle \Vert \varDelta v(\cdot ,s)\Vert _{L^2(\varOmega )}^2 ds\le \int _{\varOmega }|\nabla v(x,t)|^2 +\int ^{t+\tau }_{t}\displaystyle \Vert u(\cdot ,s)\Vert _{L^2(\varOmega )}^2 ds. $$

So (v) follows from (ii) and (iv).

Lemma 3.17

Let (a, v, w) be a classical solution of (3.4.1) with $\sigma =1$ in $(0,T_{max})$. Then there exists some $C>0$ such that

$$\begin{aligned} \Vert a(\cdot ,t)\Vert _{L^{3}(\varOmega )}\le C \end{aligned}$$

(3.4.11)

is valid for all $t\in (0,T_{max})$.

Proof

Applying Lemma 3.15 with $p=2$, one can find $k_1(A)>0$ such that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2 +\displaystyle \displaystyle \int _\varOmega e^{\xi w} |\nabla a |^2 + (2\mu -\xi \eta A )\displaystyle \int _\varOmega e^{2\xi w}a^{3}\\[4mm] \le&\displaystyle \chi ^2 \displaystyle \int _\varOmega e^{\xi w} a^2|\nabla v|^2+ k_1(A)\displaystyle \int _\varOmega e^{\xi w}a^2+ k_1(A) \displaystyle \int _\varOmega e^{\xi w} a^2 v.\\ \end{aligned} \end{aligned}$$

Therefore, by means of the Young inequality, we can get

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2 +\displaystyle \displaystyle \int _\varOmega e^{\xi w} |\nabla a |^2 +\int _\varOmega e^{\xi w} a^2 \\[4mm] \le&k_2\displaystyle \Vert a\Vert ^2_{L^4(\varOmega )} \Vert \nabla v\Vert ^2_{L^4(\varOmega )}+k_2\Vert a\Vert ^3_{L^3(\varOmega )}+k_2\Vert v\Vert ^3_{L^3(\varOmega )} +k_2,\\ \end{aligned} \end{aligned}$$

(3.4.12)

where $k_2>0$ is the constant only depending upon $A,\xi , \eta , \chi$.

On applying Lemma 3.3 with $n=2$, we have

$$ \Vert a\Vert ^2_{L^4(\varOmega )}\le k_3 \Vert \nabla a\Vert _{L^2(\varOmega )}\Vert a\Vert _{L^2(\varOmega )}+k_3\Vert a\Vert ^2_{L^2(\varOmega )} $$

and

$$ \Vert \nabla v\Vert ^2_{L^4(\varOmega )}\le k_3\Vert \triangle v\Vert _{L^2(\varOmega )}\Vert \nabla v\Vert _{L^2(\varOmega )}+k_3\Vert \nabla v\Vert ^2_{L^2(\varOmega )}, $$

which along with Lemma 3.16 (iv), implies that $\Vert \nabla v\Vert ^2_{L^4(\varOmega )}\le k_4\Vert \triangle v\Vert _{L^2(\varOmega )}+k_4.$ Hence, combining above inequalities and by the Young inequality, we have

$$\begin{aligned} k_2\displaystyle \Vert a\Vert ^2_{L^4(\varOmega )} \Vert \nabla v\Vert ^2_{L^4(\varOmega )} \le \frac{1}{2} \Vert \nabla a\Vert ^2_{L^2(\varOmega )}+k_5\Vert a\Vert ^2_{L^2(\varOmega )}(1+\Vert \triangle v\Vert ^2_{L^2(\varOmega )}). \end{aligned}$$

(3.4.13)

Therefore, inserting (3.4.13) into (3.4.12), and noting the fact $\Vert v\Vert _{L^3(\varOmega )} \le \Vert v\Vert _{W^{1,2}(\varOmega )}\le k_6$ by Lemma 3.16 (iv) (iii), we can conclude that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2 +\frac{1}{2} \displaystyle \displaystyle \int _\varOmega e^{\xi w} |\nabla a |^2 +\int _\varOmega e^{\xi w} a^2 \\ \le&k_7\displaystyle \Vert \triangle v\Vert _{L^2(\varOmega )}^2\int _\varOmega e^{\xi w}a^2+k_7 \int _\varOmega a^3+k_7. \\ \end{aligned} \end{aligned}$$

(3.4.14)

Now applying Lemma 3.3 and the Young inequality, we have

$$ \Vert a\Vert ^2 _{W^{1,2}(\varOmega )}\ge \frac{1}{\varepsilon }\Vert a\Vert ^3_{L^3(\varOmega )}-\frac{C^6_{GN}}{\varepsilon ^2}\Vert a\Vert ^4_{L^2(\varOmega )} $$

for any $\varepsilon >0$, which after inserting into (3.4.14) and taking $\varepsilon =\frac{1}{4k_7}$ says that

$$\begin{aligned} \displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2 +k_7 \displaystyle \int _\varOmega a^3 \le k_7\displaystyle \Vert \triangle v\Vert _{L^2(\varOmega )}^2\int _\varOmega e^{\xi w}a^2+k_7+k_8\Vert a\Vert ^4_{L^2(\varOmega )} \end{aligned}$$

with $k_8=16k_7^2C^6_{GN}+k_7 e^{\xi A}$.

In view of $a^3\ge \frac{1}{\varepsilon }a^2- \frac{1}{\varepsilon ^3}$ for any $\varepsilon >0$, we can obtain

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2 +\frac{1}{\varepsilon } \displaystyle \int _\varOmega e^{\xi w}a^2 \\ \le&( k_7\displaystyle \Vert \triangle v\Vert _{L^2(\varOmega )}^2+k_8\Vert a\Vert _{L^2(\varOmega )}^2) \int _\varOmega e^{\xi w}a^2+k_7+\frac{e^{3\xi A}}{\varepsilon ^3k_7^2}|\varOmega |. \\ \end{aligned} \end{aligned}$$

(3.4.15)

Now, let $\tau =\min \{1,\frac{T_{max}}{6}\}$ and $\varepsilon =\frac{\tau }{1+k_7m_3+k_8m_1}$. Then (3.4.15) implies that writing $a(t):=\frac{1}{\varepsilon }$, $b(t):=k_7\displaystyle \Vert \triangle v\Vert _{L^2(\varOmega )}^2+k_8\Vert a\Vert _{L^2(\varOmega )}^2$ and $c(t):=k_7+\frac{e^{3\xi A}}{\varepsilon ^3k_7^2}|\varOmega |$, function $y(t):=\int _\varOmega e^{\xi w}a^2$ satisfies

$$\begin{aligned} y'(t)+a(t)y(t)\le b(t)y(t)+c(t). \end{aligned}$$

(3.4.16)

Hence, the application of Lemma 3.5 to (3.4.16) with $b_1=k_7m_3+k_8m_1$, $k_1=k_7+\frac{e^{3\xi A}}{\varepsilon ^3k_7^2}|\varOmega |$ and $\rho =1$ yields

$$ \int _\varOmega e^{\xi w}a^2\le C:=e^{b_1}e^{\xi A}\Vert a_0\Vert ^2_{L^2(\varOmega )}+\frac{k_1e^{2b_1}}{1-e^{-1}}+k_1e^{b_1}. $$

Now we turn to estimate $\Vert a(\cdot ,t)\Vert _{L^{3}(\varOmega )}$. Applying the Young inequality, one obtains from (3.4.4) that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^3 +\displaystyle \frac{3}{2} \displaystyle \int _\varOmega e^{\xi w} a|\nabla a |^2 +\int _\varOmega e^{\xi w}a^3 \\ \le&6\chi ^2 \displaystyle \int _\varOmega e^{\xi w} a^3|\nabla v|^2+ 3(\xi \eta A-\mu )\displaystyle \int _\varOmega e^{2\xi w}a^{4} + 2\xi A\displaystyle \int _\varOmega e^{\xi w} a^{3}v+ 2\xi \eta A^2 \displaystyle \int _\varOmega e^{\xi w} a^{3} \\ \le&6\chi ^2 \displaystyle \int _\varOmega e^{\xi w} a^3|\nabla v|^2 +k_9 \displaystyle (\sup _{0\le t< T_{max}}\Vert v(\cdot ,t)\Vert _{L^\infty (\varOmega )}+1) \displaystyle \int _\varOmega e^{\xi w} a^{3}+k_9 \displaystyle \int _\varOmega e^{\xi w}a^{4}. \end{aligned} \end{aligned}$$

(3.4.17)

On the other hand, by $\int _\varOmega a^2\le C$ and Lemma 3.6, one can find some constant $k_{10}>0$ such that $\Vert v(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le k_{10}$ and $\Vert \nabla v(\cdot ,t)\Vert _{L^8(\varOmega )}\le k_{10}$ for all $t< T_{max}$. Hence, (3.4.17) shows that there exists $k_{11}>0$ such that

$$\begin{aligned} \displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^3 +\displaystyle \frac{2}{3} \displaystyle \int _\varOmega e^{\xi w} |\nabla a^{\frac{3}{2}} |^2 +\int _\varOmega e^{\xi w}a^3 \le k_{11} \displaystyle \int _\varOmega a^4 + k_{11}. \end{aligned}$$

(3.4.18)

By means of the Gagliardo–Nirenberg inequality, we have

$$ \Vert a\Vert _{L^4(\varOmega )}^4=\Vert a^{\frac{3}{2}}\Vert _{L^{\frac{8}{3}}(\varOmega )}^{\frac{8}{3}} \le 2 C_{c3.2-2.1}^{\frac{8}{3}}\Vert \nabla a^{\frac{3}{2}}\Vert _{L^{2}(\varOmega )}^{\frac{4}{3}} \Vert a^{\frac{3}{2}}\Vert _{L^{\frac{4}{3}}(\varOmega )}^{\frac{4}{3}}+ 2 C_1^{\frac{8}{3}} \Vert a^{\frac{3}{2}}\Vert _{L^{\frac{4}{3}}(\varOmega )}^{\frac{8}{3}}. $$

Therefore by (3.4.18), $\Vert a(\cdot ,t)\Vert _{L^2(\varOmega )}\le C$ and the Young inequality, one can arrive at

$$ \displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^3 +\int _\varOmega e^{\xi w}a^3 \le k_{12}. $$

Finally, (3.4.11) follows from the Gronwall inequality.

On applying Lemma 3.6, the following result is an immediate consequence of $0\le w(x,t)\le A$ and Lemma 3.17.

Lemma 3.18

Under the same assumptions as in Theorem 3.2, there exists $C > 0$ such that the classical solution (u, v, w) of (3.1.1) satisfies

$$\begin{aligned} \Vert v(\cdot , t)\Vert _{W^{1,\infty }(\varOmega )}\le C \quad \hbox { for all}\quad t\in (0, T_{max}). \end{aligned}$$

(3.4.19)

Note that $\Vert \nabla v(\cdot , t)\Vert _{L^\infty (\varOmega )}$ is bounded by (3.4.19). However, $\Vert \nabla w(\cdot , t)\Vert _{L^\infty (\varOmega )}$ might become unbounded. Therefore, Lemma A.1 of Tao and Winkler (2012a) as the result of the well-known Moser–Alikakos iteration Alikakos (1979) cannot be directly applied to the first equation in (3.1.1) to get the boundedness of $\Vert u(\cdot , t)\Vert _{L^\infty (\varOmega )}$. At this position, arguing as in Lemma 3.6 of Pang and Wang (2017), Lemma 4.2 of Tao (2011) or Lemma 3.5 of Tao and Winkler (2014b), we can establish the following estimates.

Lemma 3.19

Under the assumptions of Theorem 3.2, there exists $C > 0$ such that the classical solution (u, v, w) of (3.1.1) satisfies

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^\infty (\varOmega )}\le C\quad \hbox { for all}\quad t\in (0, T_{max}). \end{aligned}$$

(3.4.20)

Lemma 3.20

Under the assumptions of Theorem 3.2, for all $T>0$ there exists $C(T) > 0$ such that the classical solution (u, v, w) of (3.1.1) satisfies

$$\begin{aligned} \Vert \nabla w(\cdot ,t)\Vert _{L^5(\varOmega )}\le C(T) \quad \hbox { for all}\quad t\in (0, \min \{T, T_{max}\}). \end{aligned}$$

(3.4.21)

We are now in the position to prove Theorem 3.2 in the case $\sigma =1$.

Proof of Theorem 3.2 in the case of $\sigma =1$.  By a rather standard argument, we can show the global existence of classical solutions to (3.1.1) with $\sigma =1$, i.e., $T_{max}=+\infty$. In view of Lemma 3.18, $\Vert a(\cdot , t)\Vert _{L^\infty (\varOmega )}$ is bounded uniformly with respect to $t\in (0, T_{max})$. Combining this with Lemma 3.20, we can obtain $\Vert \nabla w(\cdot ,t)\Vert _{L^5{(\varOmega )}}\le C(T_{max})$ for all $t\in (0, T_{max})$. Hence, the statement of global existence and boundedness of classical solutions to (3.1.1) is a straightforward consequence of Lemma 3.14. Now by retracing the proof of Lemma 3.17, one can find that $\tau =1$, and thereby there exists a constant $C>0$ which is time-independent such that $\Vert a(\cdot , t)\Vert _{L^3(\varOmega )}\le C$ for all $t>0$. Therefore, $\Vert a(\cdot , t)\Vert _{L^\infty (\varOmega )}\le C$ for some $C>0$ and all $t>0$.

2. The Case of $\sigma =0$

Now we turn to proving Theorem 3.2 in the case $\sigma =0$.

Lemma 3.21

Let (u, v, w) be the classical solution of (3.1.1) with $\sigma =0$. Then we have

(i) $\Vert u(\cdot ,t)\Vert _{L^1(\varOmega )}\le m_0$   for all   $t\in (0, T_{max});$

(ii) $\displaystyle \int ^{t+\tau }_t \Vert u(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le m_1~$ for any $0<\tau \le \min \{1,\frac{ T_{max}}{3}\}$ and all $t\in (0,T_{max}-\tau )$;

(iii) $\Vert v(\cdot ,t)\Vert _{L^1(\varOmega )}\le m_0 ~~~~\text{ for } \text{ all }~~ t\in (0, T_{max});$

(iv) $\displaystyle \int ^{t+\tau }_t \Vert \nabla v(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le m_1$   for any  $0<\tau \le \min \{1,\frac{ T_{max}}{3}\}$ and all $t\in (0,T_{max}-\tau )$;

(v) $\displaystyle \int ^{t+\tau }_t \Vert \varDelta v(\cdot ,s)\Vert ^2_{L^2(\varOmega )}ds\le m_1$   for any  $0<\tau \le \min \{1,\frac{ T_{max}}{3}\}$ and all $t\in (0,T_{max}-\tau )$.

Proof

We note that we only need to show (iii), (iv) and (v) here. Integrating the elliptic equation in (3.1.1) with respect to $x\in \varOmega$ yields

$$\begin{aligned} \int _{\varOmega }v(x,t)=\int _{\varOmega }u(x,t), \end{aligned}$$

(3.4.22)

so (iii) is the consequence of (i).

Testing the equation for v in (3.1.1) by $-\varDelta v$ and integrating over $\varOmega$, we can see that

$$ \int _{\varOmega }| \varDelta v(x,t)|^2+ \int _{\varOmega }|\nabla v(x,t)|^2=-\int _{\varOmega }u \triangle v\le \frac{1}{2} \int _{\varOmega }| \varDelta v(x,t)|^2+\frac{1}{2} \int _{\varOmega } u^2(x,t) $$

and thus $\int _{\varOmega }| \varDelta v(x,t)|^2+ \int _{\varOmega }|\nabla v(x,t)|^2\le \int _{\varOmega } u^2(x,t).$ Hence (iv) and (v) follow from (ii).

As pointed out in Tao and Winkler (2014b), with the help of a well-known regularity result on semilinear second-order elliptic equations, one can only infer that $\Vert \nabla v(\cdot ,t)\Vert _{L^q(\varOmega )}\le C$ for any $1<q<2$ and all $t\in (0,T_{max})$ from Lemma 3.21 (i) and the second equation in (3.1.1) with $\sigma =0$, hence in order to allow for the choice $q=2$, some additional efforts are needed. It should be remarked that for (3.1.1) with $\sigma =1$, $\Vert \nabla v(\cdot ,t)\Vert _{L^2(\varOmega )}\le C$ can be obtained directly (see Lemma 3.16 (iv)). It is observed in Tao and Winkler (2014b) that a key step toward this is to estimate $\int _\varOmega u(\cdot ,t)\ln u(\cdot ,t)$ (see (3.16) of Tao and Winkler (2014b)). However, the estimate of the latter involves the compensation of the term $\int _\varOmega \nabla u \cdot \nabla w$, which makes the bound of $\int _\varOmega u(\cdot ,t)\ln u(\cdot ,t)$ to be time-dependent. Here, we make use of Lemmas 3.5 and 3.21 to derive the global boundedness of $\int _\varOmega u(\cdot ,t)\ln u(\cdot ,t)$.

Lemma 3.22

There exists some $C>0$ such that for any $(u_0,w_0)$ fulfilling (3.1.2), the corresponding classical solution (u, v, w) of (3.1.1) with $\sigma =0$ satisfies

$$ \int _\varOmega u(\cdot ,t)\ln u(\cdot ,t) \le C~\text{ for } \text{ all }~ t\in (0, T_{max}). $$

Proof

From the first equation in (3.4.1), it follows

$$ (ae ^{\xi w})_t=\nabla \cdot (e^{\xi w}\nabla a)-\chi \nabla \cdot ( e^{\xi w}a \nabla v)+ \mu ae^{\xi w}(r-w-ae^{\xi w}) $$

and thus

$$\begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a\ln a+ \int _\varOmega e^{\xi w}\displaystyle \frac{|\nabla a|^2}{a}+\mu \int _\varOmega a^2 e^{2\xi w}\ln a \nonumber \\ =\,&\displaystyle \int _\varOmega (e^{\xi w}a)_t\ln a+\int _\varOmega e^{\xi w}a_t+\int _\varOmega e^{\xi w}\displaystyle \frac{|\nabla a|^2}{a}+\mu \int _\varOmega a^2 e^{2\xi w}\ln a \nonumber \\ =\,&\chi \displaystyle \int _\varOmega e^{\xi w} \nabla a \cdot \nabla v+ \displaystyle \int _\varOmega ae^{\xi w}[\mu (r-w-ae^{\xi w})-\xi \eta w(1-w-ae^{\xi w})] \\&+\displaystyle \int _\varOmega ae^{\xi w}[\mu \ln a(r-w)+\xi vw] \nonumber \\ \le&\displaystyle \frac{1}{2}\int _\varOmega e^{\xi w}\displaystyle \frac{|\nabla a|^2}{a}+ \displaystyle \frac{\chi ^2}{2}\int _\varOmega e^{\xi w}a|\nabla v|^2+\frac{\mu }{4} \int _\varOmega a^2 e^{2\xi w}\ln a+k_1 \nonumber \end{aligned}$$

(3.4.23)

for some $k_1>0$ and $t\in (0, T_{max})$. Here, we may use the facts that $0\le w\le A:=\max \{\Vert w_0\Vert ,1\}$, $a^2\le \varepsilon a^2\ln a+e^{\frac{2}{\varepsilon }}$, $a\ln a\le \varepsilon a^2\ln a- \varepsilon ^{-1} \ln \varepsilon$ and $a\le \varepsilon a^2\ln a+2 e^{\frac{2}{\varepsilon }}$ for any $\varepsilon \in (0,1)$.

By Young’s inequality and applying $a^2\le \varepsilon a^2\ln a+e^{\frac{2}{\varepsilon }}$ again, we have

$$\begin{aligned} \displaystyle \frac{\chi ^2}{2}\int _\varOmega e^{\xi w}a|\nabla v|^2\le \varepsilon \int _\varOmega |\nabla v|^4+\frac{\mu }{4} \int _\varOmega a^2 e^{2\xi w}\ln a+k_2(\varepsilon ). \end{aligned}$$

(3.4.24)

Along with $a e^{\xi w}\ln a\le a^2 e^{2\xi w}\ln a+\xi A e^{2\xi A}$, combining (3.4.24) with (3.4.23) gives

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a\ln a+ \frac{1}{2}\int _\varOmega e^{\xi w}\displaystyle \frac{|\nabla a|^2}{a}+\frac{\mu }{2} \int _\varOmega a e^{\xi w}\ln a\\ \le&\displaystyle \varepsilon \int _\varOmega |\nabla v|^4+k_1+k_2(\varepsilon )+\frac{\mu \xi A}{2}e^{2\xi A}|\varOmega |, \end{aligned} \end{aligned}$$

(3.4.25)

which along with the Gagliardo–Nirenberg interpolation inequality

$$ \Vert \nabla v\Vert _{L^4(\varOmega )}^4\le C^4_{c3.2-2.1}(\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)\Vert \nabla v\Vert _{L^2(\varOmega )}^2 $$

and Lemma 3.7 entails

$$ \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a\ln a+ \frac{1}{2}\int _\varOmega e^{\xi w}\displaystyle \frac{|\nabla a|^2}{a}+\frac{\mu }{2} \int _\varOmega e^{\xi w}a\ln a\\ \le&\displaystyle \varepsilon C^4_{c3.2-2.1}(\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)\Vert \nabla v\Vert _{L^2(\varOmega )}^2 +k_3(\varepsilon )\\[1mm] \le&\displaystyle \varepsilon C^4_{c3.2-2.1}(\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2) (\alpha \int _\varOmega u\ln u+\beta )+k_3(\varepsilon ). \end{aligned} $$

Therefore, by the fact $u\ln u=e^{\xi w}a\ln a +a\xi w e^{\xi w} \le 2e^{\xi w}a\ln a +k_4$ for some $k_4>0$, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}(\int _\varOmega e^{\xi w}a\ln a+e^{\xi A}|\varOmega |)+ \frac{1}{2}\int _\varOmega e^{\xi w}\displaystyle \frac{|\nabla a|^2}{a}+\frac{\mu }{2} ( \int _\varOmega e^{\xi w}a\ln a +e^{\xi A}|\varOmega |)\\ \le&\displaystyle 2\varepsilon C^4_{c3.2-2.1}(\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2) (\alpha \int _\varOmega e^{\xi w}a\ln a +\alpha |\varOmega |k_4+\beta )+k_5(\varepsilon )\\ \le&\varepsilon k_6 (\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)(\displaystyle \int _\varOmega e^{\xi w}a\ln a +e^{\xi A}|\varOmega |)\\&+k_7( \Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)+k_5(\varepsilon ) \end{aligned} \end{aligned}$$

for $t\in (0,T_{max})$.

Now, we let the nonnegative functions a(t), b(t) and c(t) be defined by $a(t):=\frac{\mu }{2}$, $b(t):=\varepsilon k_6 (\Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)$, $c(t){:=}k_7( \Vert \varDelta v\Vert _{L^2(\varOmega )}^2+\Vert \nabla v\Vert _{L^2(\varOmega )}^2)+k_5(\varepsilon )$. Then, we see that the nonnegative function

$$ y(t):=\int _\varOmega e^{\xi w}a\ln a+e^{\xi A}|\varOmega | $$

satisfies $y'(t)+a(t)y(t)\le b(t)y(t)+c(t)$. With the help of Lemma 3.21 (iv) and (v), we can conclude that when fixing $\tau :=\min \{ 1, \frac{T_{max}}{6} \}$ and taking $\varepsilon =\frac{\mu \tau }{8k_6 m_1}$, applying Lemma 3.5 with $\rho =\frac{\mu }{4} \tau$ yields $\int _\varOmega e^{\xi w}a\ln a\le C(\tau )$ with some constant $C(\tau )>0$, which completes the proof of this lemma as $0\le w\le A$.

Based on the above $LlogL(\varOmega )$ estimate of u, we have the following.

Corollary 3.1

There exists some $C>0$ such that for any $(u_0,w_0)$ fulfilling (3.1.2), the corresponding classical solution (u, v, w) of (3.1.1) with $\sigma =0$ satisfies

$$ \Vert v(\cdot ,t)\Vert _{W^{1,2}(\varOmega )} \le C\quad ~~~~\text{ for } \text{ all }~~ t\in (0, T_{max}). $$

Proof

This follows from Lemma 3.21 (i), Lemmas 3.22 and 3.7 immediately.

At this position, we can proceed as in the proof of Lemmas 3.2–3.5 or Lemmas 3.10–3.12 of Tao and Winkler (2014b) to derive the a priori estimates below. It should be pointed out that the bounds of $\int _{\varOmega }{a\ln a}$ play an essential role in the proof of Lemma 3.11 in Tao and Winkler (2014b), while it is not necessary for our argument in the proof of Lemma 3.17 whenever $\Vert v(\cdot ,t)\Vert _{W^{1,2}(\varOmega )}$ is bounded.

Lemma 3.23

Let (u, v, w) be the classical solution of (3.1.1) with $\sigma =0$. Then there exists some $C>0$ such that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{\infty }(\varOmega )}\le C \end{aligned}$$

(3.4.26)

is valid for all $t\in (0,T_{max})$.

Proof of Theorem 3.2 in the case of $\sigma =0$.  Since the proof is very similar to that of Theorem 3.2 in the case $\sigma =1$, we omit it here.

3.4.3 Global Existence in Three-Dimensional Domains

In this subsection, inspired by Lankeit (2015); Winkler (2011b), we prove Theorem 3.3. The main idea of the proof is to verify that the quantity $\int _\varOmega a^2 (t)+\int _\varOmega |\nabla v(t)|^4$ satisfies an autonomous ordinary differential inequality, and then the comparison argument can be applied to the corresponding ordinary differential equation when $r>0$ and the initial data are suitably small.

Lemma 3.24

Let (a, v, w) be the classical solution of problem (3.4.1) in $\varOmega \times [0,T_{max})$. Then

$$\begin{aligned} \begin{aligned}&\sigma \displaystyle \frac{d}{dt}\int _\varOmega |\nabla v|^{4}+ \int _\varOmega |\nabla |\nabla v|^2|^2+4 \int _\varOmega |\nabla v|^{4} \le 7 \int _\varOmega e^{2\xi w}a^2 |\nabla v|^{2}. \end{aligned} \end{aligned}$$

(3.4.27)

Proof

We refer the interested reader to Lemma 3.2 of Tao and Winkler (2015b), (24)–(26) in Viglialoro (2017), and Lemma 4.6 in Lankeit (2015) for the proof.

Proof of Theorem 3.3 in the case of $\sigma =1$.  By Lemma 3.15, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+\displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2+(2 \mu -\xi \eta A) \displaystyle \int _\varOmega a^{3}e^{2\xi w}\\ \le&\displaystyle \chi ^2 \int _\varOmega e^{\xi w}a^2 |\nabla v|^2+ \xi A\int _\varOmega e^{\xi w}a^2 v+(2\mu r+ \xi \eta A^2) \int _\varOmega e^{\xi w}a^2. \end{aligned} \end{aligned}$$

Furthermore, the Young inequality entails that there is a constant $k_1>0$ depending upon $\xi ,\chi , \eta$ and A only such that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+\displaystyle \displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 + \int _\varOmega e^{\xi w}a^2 \\ \le&k_1 \displaystyle \int _\varOmega e^{2 \xi w}a^3+\displaystyle \int _\varOmega |\nabla v|^6+ \int _\varOmega v^3+r^2 \mu \int _\varOmega a, \end{aligned} \end{aligned}$$

which along with Lemmas 3.4 and 3.16 implies that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+\displaystyle \frac{1}{2} \displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 + \int _\varOmega e^{\xi w}a^2\\ \le&k_2\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+\displaystyle \int _\varOmega |\nabla v|^6+ \int _\varOmega v^3+r^2 \mu \int _\varOmega a+k_3(A)\left( \int _\varOmega a\right) ^3\\ \le&k_2\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+\displaystyle \int _\varOmega |\nabla v|^6+ k_4(\Vert \nabla v\Vert ^{\frac{12}{5}}_{L^2(\varOmega )}+\Vert v\Vert ^{\frac{12}{5}}_{L^1(\varOmega )}) \Vert v\Vert ^{\frac{3}{5}}_{L^1(\varOmega )} \\&+r^2 \mu \int _\varOmega a+k_3(A)\left( \int _\varOmega a\right) ^3\\ \le&k_2\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+\displaystyle \int _\varOmega |\nabla v|^6+ k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu \int _\varOmega a+k_3(A)\left( \int _\varOmega a\right) ^3. \end{aligned} \end{aligned}$$

(3.4.28)

On the other hand, from Lemmas 3.24 and 3.4, it follows that

$$ \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega |\nabla v|^{4}+ \int _\varOmega |\nabla |\nabla v|^2|^2+4 \int _\varOmega |\nabla v|^{4}\\ \le&\displaystyle \int _\varOmega a^3 + k_5(A)\int _\varOmega |\nabla v|^6\\ \le&\displaystyle \int _\varOmega a^3 + \frac{1}{2} \int _\varOmega |\nabla |\nabla v|^2|^2+k_6(A)\left( (\int _\varOmega |\nabla v|^4)^3+ (\int _\varOmega |\nabla v|^4)^{\frac{3}{2}}\right) , \end{aligned} $$

and thus

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega |\nabla v|^{4}+\frac{1}{2} \int _\varOmega |\nabla |\nabla v|^2|^2+4 \int _\varOmega |\nabla v|^{4}\\ \le&\displaystyle \int _\varOmega a^3 +k_6(A)\left( (\int _\varOmega |\nabla v|^4)^3+ (\int _\varOmega |\nabla v|^4)^{\frac{3}{2}}\right) . \end{aligned} \end{aligned}$$

(3.4.29)

Combining (3.4.29) with (3.4.28) and using Lemma 3.4 again, we can see

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}(\int _\varOmega e^{\xi w}a^2+\int _\varOmega |\nabla v|^{4})+\displaystyle \frac{1}{2} \displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 +\frac{1}{2} \int _\varOmega |\nabla |\nabla v|^2|^2+ \int _\varOmega e^{\xi w}a^2 \\&+4 \int _\varOmega |\nabla v|^{4}\\ \le&\displaystyle \int _\varOmega a^3+\displaystyle \int _\varOmega |\nabla v|^6+ k_2\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+k_6(A)\left( (\int _\varOmega |\nabla v|^4)^3+ (\int _\varOmega |\nabla v|^4)^{\frac{3}{2}}\right) \\&+ k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu \displaystyle \int _\varOmega a+k_3(A)(\displaystyle \int _\varOmega a)^3\\ \le&\displaystyle \frac{1}{4} \displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 +\frac{1}{4} \int _\varOmega |\nabla |\nabla v|^2|^2+ k_7\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3 \\&+k_8(A)\left( (\int _\varOmega |\nabla v|^4)^3+ (\int _\varOmega |\nabla v|^4)^{\frac{3}{2}}\right) \\&+ k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu \displaystyle \int _\varOmega a+k_9(A)(\displaystyle \int _\varOmega a)^3. \end{aligned} \end{aligned}$$

(3.4.30)

Finally, by the Young inequality, we obtain that

$$ k_8(A) (\int _\varOmega |\nabla v|^4)^{\frac{3}{2}} \le 2 \int _\varOmega |\nabla v|^4+ k_{10}(A)(\int _\varOmega |\nabla v|^4)^3. $$

Hence, applying Lemma 3.16, (3.4.30) shows that $y(t):=\int _\varOmega e^{\xi w}a^2+\int _\varOmega |\nabla v|^{4}$, $t>0$, satisfies

$$ \begin{aligned} y'(t)+y(t)\le&k_{11}(A)y^3 (t)+ k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu \displaystyle \int _\varOmega a (\cdot ,t) +k_9(A)(\displaystyle \int _\varOmega a( \cdot ,t))^3\\ \le&k_{11}(A)y^3 (t)+ k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu m_0+k_9(A)m_0^3\\ \end{aligned} $$

for some $k_{11}(A)>0$.

Now we can conclude that there is a positive constant $r_0$ such that function $\Theta (\varsigma ):=-\varsigma +k_{11}(A)\varsigma ^3 +k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu m_0+k_9(A)m_0^3$, $\varsigma \ge 0$, attains its minimum at $\varsigma _0=(\frac{1}{3k_{11}(A)})^{\frac{1}{2}}$, and $\Theta (\varsigma _0)<0$ when $r<r_0$, and $\Vert u_0\Vert _{L^1(\varOmega )}$ and $\Vert v_0\Vert _{W^{1,2}(\varOmega )}$ are suitably small. In fact, it is observed that $\Theta (\varsigma _0)<0$ provided that $k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) +r^2 \mu m_0+k_9(A)m_0^3 <\frac{2\varsigma _0}{3}$. To this end, taking

$$r_0=\min \{1,\displaystyle \frac{1}{ |\varOmega |},\frac{\varsigma _0}{ 2|\varOmega |}(\mu +c_{9}(A)+2c_{4}(1+\frac{1}{\mu } ))^{-1} \}$$

and by continuity of the expressions $m_0$, $m_2$ and $\tilde{m}_0$, one can verify that $k_4\tilde{m}_0^{\frac{3}{5}} (m_2^{\frac{6}{5}}+\tilde{m}_0^{\frac{12}{5}}) {+}r^2 \mu m_0{+}k_9(A)m_0^3 {<}\frac{2\varsigma _0}{3}$ is indeed valid if $r{<}r_0$, and $\Vert u_0\Vert _{L^1(\varOmega )}$ and $\Vert v_0\Vert _{W^{1,2}(\varOmega )}$ are suitably small. The comparison principle for ordinary differential equations $y'(t)\le \Theta (y(t))$ therefore shows by means of comparison with $y\equiv \varsigma _0$ that $y(t) \le \varsigma _0$ for all $t \ge 0$ when $y(0)\le \varsigma _0$, which can be satisfied whenever $\Vert u_0\Vert _{L^2(\varOmega )}$ and $\Vert v_0\Vert _{W^{1,4}(\varOmega )}$ are sufficiently small.

The next step is to obtain a bound for a with respect to the norm in $L^\infty (\varOmega )$ by a bootstrap procedure, on the basis of the bounds on $\Vert a\Vert _{L^2(\varOmega )}$ and $\Vert v\Vert _{W^{1,4}(\varOmega )}$.

By Lemma 3.15 with $p=3$, we get

$$ \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^3+\displaystyle \frac{4}{3}\int _\varOmega e^{\xi w}|\nabla a^{\frac{3}{2}} |^2+3(\mu -\xi \eta A) \displaystyle \int _\varOmega a^{4}e^{2\xi w}\\ \le&\displaystyle 3\chi ^2 \int _\varOmega e^{\xi w}a^3 |\nabla v|^2+2 \xi A\int _\varOmega e^{\xi w}a^3 v+(3\mu r+ 2\xi \eta A^2) \int _\varOmega e^{\xi w}a^3. \end{aligned} $$

Moreover, due to $W^{1,4}(\varOmega )\hookrightarrow L^\infty (\varOmega )$ and applying Lemma 3.3, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^3+\displaystyle \int _\varOmega |\nabla a^{\frac{3}{2}} |^2+\int _\varOmega e^{\xi w}a^3\\ \le&\displaystyle 3\chi ^2 e^{\xi A} \int _\varOmega a^3 |\nabla v|^2+ \int _\varOmega a^4 +k_{12}(A)\\ \le&3\chi ^2e^{\xi A} (\displaystyle \int _\varOmega a^6)^{\frac{1}{2}} (\displaystyle \int _\varOmega |\nabla v|^4)^{\frac{1}{2}}+ \int _\varOmega a^4 +k_{12}(A)\\ \le&k_{13}(A)\left( (\displaystyle \int _\varOmega |\nabla a^{\frac{3}{2}} |^2)^{\frac{6}{7}}(\displaystyle \int _\varOmega a^2)^{\frac{3}{14}} +(\displaystyle \int _\varOmega a^2)^{\frac{3}{2}}\right) (\displaystyle \int _\varOmega |\nabla v|^4)^{\frac{1}{2}}\\&+(\displaystyle \int _\varOmega |\nabla a^{\frac{3}{2}} |^2)^{\frac{6}{7}}(\displaystyle \int _\varOmega a^2)^{\frac{5}{7}}+ (\displaystyle \int _\varOmega a^2)^2 +k_{12}(A)\\ \le&\displaystyle \frac{1}{2} \int _\varOmega |\nabla a^{\frac{3}{2}} |^2+k_{14}(A), \end{aligned} \end{aligned}$$

which implies that $\int _\varOmega e^{\xi w}a^3\le C$ for some $C>0$. At this position, similarly as in the proof of Theorem 3.2, one can derive $\Vert a(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C$ and $\Vert v(\cdot ,t)\Vert _{W^{1,\infty }(\varOmega )}\le C$ for some time-independent constant $C>0$ and all $t>0$, and then $\Vert \nabla w(\cdot ,t)\Vert _{L^5{(\varOmega )}}\le C(T)$ for all $t\in (0, T)$, which along with Lemma 3.14 completes the proof of this theorem.

Proof of Theorem 3.3 in the case of $\sigma =0$.  Since the proof is similar to that of the case $\sigma =1$, we may confine ourselves to an outline, giving only details in places which are characteristic for the present setting.

By Lemma 3.15 and Young’s inequality, we have

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+\displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 + 2\int _\varOmega e^{\xi w}a^2\\ \le&k_1(A) \int _\varOmega a^3+\displaystyle \int _\varOmega |\nabla v|^6+ \int _\varOmega v^3+r^2 \mu \int _\varOmega a. \end{aligned} \end{aligned}$$

(3.4.31)

On the other hand, from Lemma 3.24, it follows that

$$\begin{aligned} \displaystyle \int _\varOmega |\nabla |\nabla v|^2|^2+4 \int _\varOmega |\nabla v|^{4} \le k_2(A) \displaystyle \int _\varOmega a^2 |\nabla v|^{2} \le \displaystyle k_3(A) \int _\varOmega a^3+\displaystyle \int _\varOmega |\nabla v|^6. \end{aligned}$$

(3.4.32)

Combining (3.4.32) with (3.4.31) and applying Lemma 3.4, we can obtain

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+\displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 + 2\int _\varOmega e^{\xi w}a^2+\displaystyle \int _\varOmega |\nabla |\nabla v|^2|^2+4 \int _\varOmega |\nabla v|^{4}\\ \le&\displaystyle k_4(A) \int _\varOmega a^3+2\displaystyle \int _\varOmega |\nabla v|^6+ \int _\varOmega v^3+r^2 \mu \int _\varOmega a\\ \le&\displaystyle \frac{1}{2} \displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 +\frac{1}{2} \int _\varOmega |\nabla |\nabla v|^2|^2+ k_5(A)\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+k_6(A)(\displaystyle \int _\varOmega a)^3\\&+r^2\displaystyle \mu \int _\varOmega a +k_7\displaystyle (\int _\varOmega |\nabla v|^4)^3+ \int _\varOmega |\nabla v|^4+ \int _\varOmega v^3\\ \le&\displaystyle \frac{1}{2} \displaystyle \int _\varOmega e^{\xi w}|\nabla a |^2 +\frac{1}{2} \int _\varOmega |\nabla |\nabla v|^2|^2+ k_5(A)\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+k_6(A)(\displaystyle \int _\varOmega a)^3\\&+r^2\displaystyle \mu \int _\varOmega a +k_7\displaystyle (\int _\varOmega |\nabla v|^4)^3+ 2 \int _\varOmega |\nabla v|^4+k_8 (\int _\varOmega v)^3. \end{aligned} \end{aligned}$$

(3.4.33)

By the Gagliardo–Nirenberg inequality,

$$\begin{aligned} \Vert v\Vert ^3_{L^3{(\varOmega )}}&\le C^3_{c3.2-2.1}\Vert \nabla v\Vert ^{\frac{24}{13}}_{L^4{(\varOmega )}}\Vert v\Vert ^{\frac{15}{13}}_{L^1{(\varOmega )}}+k_9\Vert v\Vert ^3_{L^1{(\varOmega )}}\\&\le \Vert \nabla v\Vert ^{4}_{L^4{(\varOmega )}}+k_{10}\Vert v\Vert ^3_{L^1{(\varOmega )}}. \end{aligned} $$

Hence from (3.4.33), it follows that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+ 2\int _\varOmega e^{\xi w}a^2+2 \int _\varOmega |\nabla v|^{4}\\ \le&k_5(A)\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+k_6(A)(\displaystyle \int _\varOmega a)^3 +r^2\displaystyle \mu \int _\varOmega a +k_7\displaystyle (\int _\varOmega |\nabla v|^4)^3+k_8 (\int _\varOmega v)^3. \end{aligned} \end{aligned}$$

(3.4.34)

On the other hand, applying the standard elliptic regularity theory in the three-dimensional setting to the second equation in (3.4.1), we have

$$ \displaystyle \int _\varOmega |\nabla v|^4\le \Vert v\Vert ^4_{W^{2,2}(\varOmega )}\le k_{11} \Vert e^{\xi w}a\Vert ^4_{L^{2}(\varOmega )} $$

for some $k_{11}>0$, which combined with (3.4.34) and the Young inequality says that

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _\varOmega e^{\xi w}a^2+ 2\int _\varOmega e^{\xi w}a^2\\ \le&k_5(A)\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^3+k_6(A)(\displaystyle \int _\varOmega a)^3 +r^2\displaystyle \mu \int _\varOmega a + k_{12}(A)\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^6+k_8 (\int _\varOmega v)^3\\ \le&\displaystyle \int _\varOmega e^{\xi w}a^2+ k_{13}(A)\displaystyle \left( \int _\varOmega e^{\xi w}a^2\right) ^6+k_{14}(A)(\displaystyle \int _\varOmega a)^3 +r^2\displaystyle \mu \int _\varOmega a, \end{aligned} \end{aligned}$$

(3.4.35)

where we have used the fact $\int _\varOmega v=\int _\varOmega u$. Therefore along with Lemma 3.21 (i), (3.4.35) shows that $y(t):=\int _\varOmega e^{\xi w}a^2$, $t>0$, satisfies

$$ \begin{aligned} y'(t)+y(t)&\le k_{13}(A)y^6(t)+k_{14}(A)(\displaystyle \int _\varOmega a)^3 +r^2\displaystyle \mu \int _\varOmega a\\[1mm]&\le k_{13}(A)y^6(t)+k_{14}(A)m_0^3 +r^2\displaystyle \mu m_0. \end{aligned} $$

At this point, the proof can be completed by arguments similar to those for the case $\sigma =1$.

3.5 Asymptotic Behavior of Solutions to a Chemotaxis–Haptotaxis Model

3.5.1 Global Boundedness

In this part, we first recall the result on local existence and uniqueness of classical solutions to (3.1.4) as well as a convenient extensibility criterion, which follows from Theorem 3.1, Lemma 5.9 and Theorem 5.1 of Morales-Rodrigo and Tello (2014).

Lemma 3.25

(Morales-Rodrigo and Tello (2014)) Let $\varOmega \subset \mathbb {R}^n$ be a smooth bounded domain. There exists $T_{max}\in (0, \infty ]$ such that the problem (3.1.4) possesses a unique classical solution satisfying $(p,c,w)\in (C(\bar{\varOmega }\times [0,T_{\max }))\cap C^{2,1}(\bar{\varOmega }\times (0,T_{\max }))^3$. Moreover, for any $s>n+2$,

$$\begin{aligned} \limsup _{t\nearrow T_{\max }}\Vert p(\cdot ,t)\Vert _{W^{1,s}(\varOmega )}\rightarrow \infty \end{aligned}$$

(3.5.1)

if $T_{\max }<+\infty$.

From now on, let (p, c, w) be the local classical solution of (3.1.4) on $(0,T_{\max })$ provided by Lemma 3.25, and $\tau :=\min \{1,\frac{T_{\max }}{6}\}$.

The following basic but important properties of the solution to (3.1.4) can be directly obtained via standard arguments.

Lemma 3.26

(Morales-Rodrigo and Tello (2014)) There exists a positive constant C independent of time such that

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\varOmega } p(\cdot ,t)\le C_1:=\max \{\int _\varOmega p_0,|\varOmega |\},~ \displaystyle \int ^{t}_{0}e^{-2s} \int _{\varOmega }p^2ds \le C~~\text{ for } \text{ all }~~t\in (0, T_{\max }), \end{aligned} \end{aligned}$$

(3.5.2)

$$\begin{aligned} \displaystyle \int ^{t+\tau }_{t}\int _{\varOmega } p^2\le C_1(1+\frac{1}{\lambda })~~~\hbox {for all}~~t\in (0, T_{\max }-\tau ), \end{aligned}$$

(3.5.3)

$$\begin{aligned} c(t)\le \Vert c_0\Vert _{L^\infty (\varOmega )}e^{-t},~ \displaystyle \int _\varOmega |\nabla c(t)|^2+ \int ^{t}_{0}\int _{\varOmega }(|\nabla c|^2+ |\varDelta c|^2)\le C ~~\text{ for } \text{ all }~~t\in (0, T_{\max }), \end{aligned}$$

(3.5.4)

$$\begin{aligned} 0\le w(t)\le \max \{\Vert w_0\Vert _{L^\infty (\varOmega )},1\}~~\text{ for } \text{ all }~~t\in (0, T_{\max }).\end{aligned}$$

(3.5.5)

As the proof of Theorem 3.4 in the one-dimensional case is similar to that for two dimensions, henceforth in this section, we shall focus on the case $n=2$.

First, we shall show that p remains bounded in $L^q(\varOmega )$ for any finite q. We note that the $L^ q(\varOmega )$-bound in Lemma 3.10 of Morales-Rodrigo and Tello (2014) depends on the time variable.

Lemma 3.27

For any $r\in (1, \infty )$, there exists a positive constant $C(r,\tau )$ independent of t, such that $\Vert p(\cdot ,t)\Vert _{L^r(\varOmega )}\le C(r,\tau )$ for all $t\in (0, T_{\max })$.

Proof

Let $q := p(c + 1)^{-\alpha } e^{-\rho w}$. As in the proof of Lemma 3.10 in Morales-Rodrigo and Tello (2014), we infer that for any $m=1,2, \ldots$ there exist constants $c(m)>0$ depending upon m and $C_1>0$ such that

$$\begin{aligned}&\frac{d}{dt}\int _{\varOmega }q^{2^m}(c+1)^\alpha e^{\rho w}+\int _{\varOmega } |\nabla q^{2^{m-1}} |^2 \\ \le&c(m)(\int _{\varOmega } |\varDelta c |^2+1)\int _{\varOmega }q^{2^m}+ c(m)(\int _{\varOmega }q^{2^m})^2+C_1.\nonumber \end{aligned}$$

(3.5.6)

Next, we use induction to show

$$\begin{aligned} \int _{\varOmega }q^{2^m}+\int ^{t+\tau }_t\int _{\varOmega } |\nabla q^{2^{m-1}} |^2\le C(m). \end{aligned}$$

(3.5.7)

Taking $m = 1$ in (3.5.6), we get

$$\begin{aligned}&\frac{d}{dt}\int _{\varOmega }q^{2}(c+1)^\alpha e^{\rho w}+\int _{\varOmega } |\nabla q |^2 \\ \le&c(1)(\displaystyle \int _{\varOmega } |\varDelta c |^2+1 +\int _{\varOmega }q^{2})\int _{\varOmega }q^2(c+1)^\alpha e^{\rho w}+C_1,\nonumber \end{aligned}$$

(3.5.8)

which implies that for the functions

$$y(t)=\int _{\varOmega }q^{2}(c+1)^\alpha e^{\rho w}\quad \hbox {and}\quad a(t)= c(1)(\int _{\varOmega } |\varDelta c |^2+1 +\int _{\varOmega }q^{2}),$$

we have $\frac{d y}{dt}\le a(t)y +C_1.$ On the other hand, for any given $t>\tau$, it follows from (3.5.2) that there exists some $t_0\in [t-\tau ,t]$ such that $y(t_0)\le \frac{C_1}{\tau }(1+\frac{1}{\lambda })$. Hence, by ODE comparison argument we get

$$\begin{aligned} y(t)\le y(t_0)e^{\int ^t_{t_0}a(s)ds} +C_1\int ^t_{t_0}e^{\int ^t_{s}a(\tau )d\tau } ds\le C_2. \end{aligned}$$

(3.5.9)

In this inequality, we have taken $t_0=0$ if $t\le \tau$ and noticed that $\int ^{t}_{t-\tau } a(s)ds\le C_3$ for all $t<T_{\max }$ by Lemma 3.25. Combining (3.5.8) with (3.5.9), one can see that (3.5.7) is indeed valid for $m=1$.

Now, suppose that (3.5.7) is valid for an integer $m+1=k\ge 2$, i.e.,

$$\begin{aligned} \int _{\varOmega }q^{2^{k-1}}+\int ^{t+\tau }_t\int _{\varOmega } |\nabla q^{2^{k-2}} |^2\le C(k). \end{aligned}$$

(3.5.10)

By the Gagliardo–Nirenberg inequality in two dimensions

$$\Vert z\Vert ^4_{L^4(\varOmega )}\le C_3 \Vert \nabla z\Vert ^2_{L^2(\varOmega )}\Vert z\Vert ^2_{L^2(\varOmega )}+C_4 \Vert z\Vert ^4_{L^2(\varOmega )}, $$

and hence

$$\begin{aligned} \int _{\varOmega }q^{2^{k}}\le C_3 \int _{\varOmega } |\nabla q^{2^{k-2}} |^2 \int _{\varOmega }q^{2^{k-1}}+C_4 (\int _{\varOmega }q^{2^{k-1}} )^2. \end{aligned}$$

(3.5.11)

Integrating (3.5.10) between t and $t+\tau$, we have

$$\begin{aligned} \int ^{t+\tau }_t\int _{\varOmega }q^{2^{k}}\le C_5(k), \end{aligned}$$

which implies that for any $t{\ge } \tau$, there exists some $t_0\in [t-\tau ,t]$ such that $\int _{\varOmega }q^{2^{k}} (t_0)\le C_6$. At this point, let

$$y(t):=\int _{\varOmega }q^{2^k}(c+1)^\alpha e^{\rho w}\quad \hbox {and}\quad b(t)= c(k)(\int _{\varOmega } |\varDelta c |^2+1 +\int _{\varOmega }q^{2^k}).$$

Then (3.5.6) can be rewritten as

$$\frac{d y}{dt}+\int _{\varOmega } |\nabla q^{2^{k-1}} |^2\le b(t)y +C_1.$$

By the argument above, one can obtain

$$\begin{aligned} \int _{\varOmega }q^{2^{k}}+\int ^{t+\tau }_t\int _{\varOmega } |\nabla q^{2^{k-1}} |^2\le C, \end{aligned}$$

and thereby conclude that (3.5.7) is valid for all integers $m\ge 1$. The proof of Lemma 3.27 is now complete in view of the boundedness of the weight $(c+1)^\alpha e^{\rho w}$.

To establish a priori estimates of $\Vert p(\cdot ,t)\Vert _{L^\infty (\varOmega )}$, we need some fundamental estimates for the solution of the following problem:

$$\begin{aligned} \left\{ \begin{aligned}&c_t=\varDelta c -c+f,&x\in \varOmega , t>0,\\&\displaystyle \frac{\partial c}{\partial \nu }=0,&x\in \partial \varOmega , t>0,\\&c(x,0)=c_0(x),&x\in \varOmega .\\ \end{aligned}\right. \end{aligned}$$

(3.5.12)

Lemma 3.28

([Li and Wang 2018, Lemma 2.2]) Let $T>0$, $r\in (1,\infty )$. Then for each $c_0\in W^{2,r}(\varOmega )$ with $\displaystyle \frac{\partial c_0}{\partial \nu }=0$ on $\partial \varOmega$ and $f\in L^r(0,T;L^r(\varOmega ))$, (3.5.12) has a unique solution $c\in W^{1,r}(0,T; L^r(\varOmega ))\cap L^r(0,T; W^{2,r}(\varOmega ))$ given by

$$ c(t)=e^{-t}e^{t\varDelta }c_0+\int ^t_0 e^{-(t-s)}e^{(t-s)\varDelta }f(s)ds, \quad t\in [0,T], $$

where $e^{t\varDelta }$ is the semigroup generated by the Neumann Laplacian, and there is $C_r>0$ such that

$$\begin{aligned} \int ^t_0\int _\varOmega e^{rs}|\varDelta c(x,s)|^r dxds\le C_r \int ^t_0\int _\varOmega e^{rs} |f(x,s)|^rdxds + C_r\Vert v_0\Vert _{W^{2,r}(\varOmega )}. \end{aligned}$$

Now applying these estimates to control the cross-diffusive flux appropriately, we can derive the boundedness of p in $\varOmega \times (0, T_{max})$.

Lemma 3.29

There exists a constant $C>0$ independent of t such that $\Vert p(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C$ for all $t\in (0, T_{max})$.

Proof

We will only give a sketch of the proof, which is similar to that of Lemma c3.3–3.13 of Morales-Rodrigo and Tello (2014). For $k\ge \max \{ 2,\Vert p_0\Vert _{L^\infty (\varOmega )}\}$, let $q_k=\max \{q-k,0\}$ and $\varOmega _k(t)=\{x\in \varOmega : q(x,t)>k \}$. Multiplying the equation of q by $q_k$, we obtain

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}+2\int _{\varOmega } |\nabla q_k |^2+2\int _{\varOmega } q_k ^2+8\int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}\\ \le&\displaystyle C_1\int _{\varOmega } q_k ^3+ C_1k\displaystyle \int _{\varOmega } q_k^2 +C_1k^2 \int _{\varOmega } q_k+ C_1 \displaystyle \int _{\varOmega }(q_k^2+kq_k) |\varDelta c | \end{aligned} \end{aligned}$$

(3.5.13)

for some $C_1>0$ independent of k. By the boundedness of q in $L^r(\varOmega )$ for any $r>1$, the Gagliardo–Nirenberg inequality and Young inequality, we obtain

$$\displaystyle C_1\Vert q_k\Vert ^3_{L^3(\varOmega )}\le \frac{1}{4}\Vert q_k\Vert ^2_{H^1(\varOmega )} +C_2\Vert q_k\Vert _{L^1(\varOmega )}, $$

$$ \displaystyle C_1k\Vert q_k\Vert ^2_{L^2(\varOmega )}\le \frac{1}{4}\Vert q_k\Vert ^2_{H^1(\varOmega )} +C_2k^2\Vert q_k\Vert _{L^1(\varOmega )},$$

$$C_1 \displaystyle \int _{\varOmega }q_k^2 |\varDelta c |\le \frac{1}{4}\Vert q_k\Vert ^2_{H^1(\varOmega )} +C_2\Vert q_k\Vert ^2_{L^2(\varOmega )}\Vert \varDelta c\Vert ^2_{L^2(\varOmega )},$$

$$C_1 k\displaystyle \int _{\varOmega }q_k |\varDelta c |\le \frac{1}{4}\Vert q_k\Vert ^2_{H^1(\varOmega )} +C_2k^2(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )})|\varOmega _k|^{\frac{3}{2}}.$$

Inserting the above estimates into (3.5.10), we have

$$ \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}+\int _{\varOmega } |\nabla q_k |^2+\int _{\varOmega } q_k ^2+8\int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}\\ \le&C_2\Vert q_k\Vert ^2_{L^2(\varOmega )}\Vert \varDelta c\Vert ^2_{L^2(\varOmega )}+C_2k^2(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )})|\varOmega _k|^{\frac{3}{2}} +(C_1+2C_2)k^2\Vert q_k\Vert _{L^1(\varOmega )}\\ \le&C_2\Vert q_k\Vert ^2_{L^2(\varOmega )}\Vert \varDelta c\Vert ^2_{L^2(\varOmega )}+\displaystyle \frac{1}{2}\Vert q_k\Vert ^2_{H^1(\varOmega )}+C_3k^4(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )})|\varOmega _k|^{\frac{3}{2}}. \end{aligned} $$

On the other hand, according to the relation between distribution functions and $L^p$ integrals (see, e.g., (2.6) of Reyes and Vázquez (2006)), we can see that

$$ (r+1)\int ^\infty _0 s^r|\varOmega _s(t)|ds=\Vert q(t)\Vert ^{r+1}_{L^{r+1}(\varOmega )}. $$

Hence, taking into account Lemma 3.26, we get

$$(k-1)^{16}|\varOmega _k(t)|< \int ^k_{k-1} s^{16}|\varOmega _s(t)|ds<\int ^\infty _0 s^{16}|\varOmega _s(t)|ds\le \frac{1}{17}\Vert q(\cdot ,t)\Vert ^{17}_{L^{17}(\varOmega )}$$

and thus

$$\begin{aligned} \begin{aligned}&\displaystyle \frac{d}{dt}\int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}+8\int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}\\ \le&C_2\Vert \varDelta c\Vert ^2_{L^2(\varOmega )}\displaystyle \int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}+C_4(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )})|\varOmega _k|^{\frac{5}{4}}. \end{aligned} \end{aligned}$$

Therefore, if $h(t)=8-C_2\Vert \varDelta c\Vert ^2_{L^2(\varOmega )}$, then

$$ \int _{\varOmega }q_k^{2}(c+1)^\alpha e^{\rho w}\le C_4 e^{-\int ^t_0 h(s)ds}\int ^t_0(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )})e^{\int ^s_0 h(\sigma )d\sigma } |\varOmega _k(s)|^{\frac{5}{4}}ds. $$

Furthermore, since $e^{-\int ^t_0 h(s)ds}=e^{-8t} e^{C_2\int ^t_0 \Vert \varDelta c\Vert ^2_{L^2(\varOmega )}ds} \le C_5 e^{-8t}$ by Lemma 3.25 and $e^{\int ^s_0 h(\sigma )d\sigma }\le e^{8s}$, we get

$$ \begin{aligned} \displaystyle \int _{\varOmega }q_k^{2}\le&C_6 \displaystyle \int ^t_0 e^{-8(t-s)}(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )}) |\varOmega _k(s)|^{\frac{5}{4}}ds\\ \le&C_6 \displaystyle \int ^t_0 e^{-8(t-s)}(1+\Vert \varDelta c\Vert ^8_{L^8(\varOmega )})ds\cdot \displaystyle \sup _{t\ge 0}|\varOmega _k(t)|^{\frac{5}{4}}. \end{aligned} $$

To estimate the integral term in the right-hand side of the above inequality, we apply Lemma 3.28 with $r=8$ and Lemma 3.27 to get

$$ \int ^t_0 e^{-8(t-s)}\Vert \varDelta c\Vert ^8_{L^8(\varOmega )}ds\le C_7 $$

and thus $\int _{\varOmega }q_k^{2}\le C_8 ( \displaystyle \sup _{t\ge 0}|\varOmega _k(t)| )^{\frac{5}{4}}$.

On the other hand, $\int _{\varOmega }q_k^{2}(t)\ge \int _{\varOmega _j(t)}q_k^{2}(t)\ge (j-k)^2|\varOmega _j(t)|$ for $j>k$. Consequently

$$ (j-k)^2\displaystyle \sup _{t\ge 0}|\varOmega _j(t)|\le C_8 ( \displaystyle \sup _{t\ge 0}|\varOmega _k(t)| )^{\frac{5}{4}} |\varOmega _k(t)|. $$

According to Lemma B.1 of Kinderlehrer and Stampacchia (1980), there exists $k_0<\infty$ such that $|\varOmega _{k_0}(t)|=0$ for all $t\in (0, T_{max})$. Therefore, $\Vert q(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le k_0$ for any $t\in (0, T_{max})$ and thereby the proof is complete.

Proof of Theorem 3.4.  By the boundedness of p in $L^\infty ((0, T_{max}),L^\infty (\varOmega ))$ from Lemma 3.29 and a bootstrap argument as in Morales-Rodrigo and Tello (2014), we can see that the global existence of classical solutions to (3.1.4) is an immediate consequence of Lemma 3.25, i.e., $T_{\max } = \infty$. Indeed, suppose that $T_{\max }<\infty$, then by Lemmas 3.15 and 3.19 of Morales-Rodrigo and Tello (2014), we can see that for any $s>n+2$ and $t\le T_{\max }$

$$\Vert c(\cdot ,t)\Vert _{W^{1,s}(\varOmega )}+\Vert w(\cdot ,t)\Vert _{W^{1,s}(\varOmega )}\le C. $$

Further by Lemma 3.20 of Morales-Rodrigo and Tello (2014), we have $\Vert p(\cdot ,t)\Vert _{W^{1,s}(\varOmega )}\le C$ which contradicts (3.5.1) and thus implies that $T_{\max } = \infty$. Moreover, since $\tau :=\min \{1,\frac{T_{\max }}{6}\}=1$, there exists a constant $C>0$ independent of time t such that $\Vert p(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C$ for all $t\ge 0$ by retracing the proofs of Lemmas 3.27 and 3.28. This completes the proof of Theorem 3.4.

3.5.2 Asymptotic Behavior

In this part, on the basis of the $L^\infty$-bound of p provided by Theorem 3.4, we shall look at the asymptotic behavior of the solution (p, c, w) of the problem (3.1.4).

1. $L^r$ -convergence of solutions in two dimensions

When either $w_0 > 1$ or $\Vert w_0-1\Vert _{L^\infty (\varOmega )}<\delta$ for some $\delta > 0$, the authors of Morales-Rodrigo and Tello (2014) removed the time dependence of the $L^\infty$-bound of p (see Lemma 5.8 of Morales-Rodrigo and Tello (2014)) and thereby investigated the asymptotic behavior of solutions to (3.1.4). In this subsection, on the basis of the $L^\infty$-bound of p being independent of time as provided by Theorem 3.4, we shall derive the same estimates as in Lemmas 5.6 and 5.7 of Morales-Rodrigo and Tello (2014) under the weaker assumption that $w_0>1-\frac{1}{\rho }$. We shall show that the solution (p, c, w) to (3.1.4) converges to the homogeneous steady state (1, 0, 1) as $t\rightarrow \infty$.

Before going into the details, let us first collect some useful related estimates. It should be noted that no other assumptions on the initial data $(p_0, c_0, w_0)$ are made except for reasonable regularity, i.e., (3.1.5).

Lemma 3.30

([Morales-Rodrigo and Tello 2014, Lemmas 3.4, 3.8, 5.1, 5.2]) Let (p, c, w) be the global, classical solution of (3.1.4). Then

$$\begin{aligned} \left| \displaystyle \int _ 0^\infty \int _{\varOmega } p(1-p)\right| \le \max \{\int _\varOmega p_0,|\varOmega |\}/\lambda ; \end{aligned}$$

(3.5.14)

$$\begin{aligned} \displaystyle \int _ 0^\infty \int _{\varOmega } p|w-1| \le \Vert w_0-1\Vert _{L^1(\varOmega )}; \end{aligned}$$

(3.5.15)

$$\begin{aligned} \displaystyle \int _ 0^\infty \int _{\varOmega } p|\nabla c|^2 < \infty ; \end{aligned}$$

(3.5.16)

$$\begin{aligned} \displaystyle \int _{\varOmega } |\nabla c(t)|^2\le e^{-2t}\left( \int _{\varOmega } |\nabla c_0|^2+\mu ^2\Vert c_0\Vert ^2_{L^\infty (\varOmega )}\max \{\int _\varOmega p_0,|\varOmega |\}(t+\frac{1}{\lambda })\right) . \end{aligned}$$

(3.5.17)

Lemma 3.31

Under the assumptions of Theorem 3.4, we have

$$\begin{aligned} \displaystyle \sup _{t\ge 0}\Vert c(t)\Vert _{W^{1,\infty }(\varOmega )} \le C. \end{aligned}$$

(3.5.18)

Proof

We know that c solves the linear equation

$$ c_t=\varDelta c-c+f $$

under the Neumann boundary condition with $f := -\mu pc$. Since $p\ge 0$, we know that $0\le c(x,t)\le$ $\Vert c_0\Vert _{L^\infty (\varOmega )} e^{-t}$ by the standard sub-super solutions method. On the other hand, by Theorem 3.4, $\displaystyle \sup _{t\ge 0}$ $\Vert p(t)\Vert _{L^\infty (\varOmega )}$ $\le C_1$, which readily implies that $\displaystyle \Vert f\Vert _{L^\infty ((0, \infty );L^\infty (\varOmega ))}\le C_1$. Now upon a standard regularity argument, we can deduce the desired result. For the reader’s convenience, we only give a brief sketch of the main ideas, and would like to refer to the proof of Lemma 1 in Kowalczyk and Szymańska (2008) or Lemma 4.1 in Horstmann and Winkler (2005) for more details. Indeed, according to the variation-of-constants formula of c, we have for $t>2$

$$ c(\cdot ,t)= e^{(t-1)(\varDelta -1)}c(\cdot ,1)+\int _{1}^t e^{(t-s)(\varDelta -1)}f(\cdot ,s)ds. $$

So by Lemma 1.1 (ii), we infer that

$$\begin{aligned} \begin{aligned}&\Vert \nabla c(\cdot ,t)\Vert _{L^\infty (\varOmega )} \\\displaystyle \le&2c_2 \Vert c(\cdot ,1)\Vert _{L^1(\varOmega )}+c_2\int _{1}^t (1+(t-s)^{-\frac{1}{2}}) e^{-(t-s)(\lambda _1+1)}\Vert f(\cdot ,s)\Vert _{L^\infty (\varOmega )}ds\\ \le&2c_2 \Vert c(\cdot ,1)\Vert _{L^1(\varOmega )}+c_2C_1\displaystyle \int _{0}^\infty (1+\sigma ^{-\frac{1}{2}}) e^{-\sigma }d\sigma . \end{aligned} \end{aligned}$$

Lemma 3.32

([Morales-Rodrigo and Tello 2014, Lemma 5.4]) Let (p, c, w) be the global, classical solution of (3.1.4). Then for every $t\ge 0$ and $\kappa >0$,

$$\begin{aligned} \displaystyle \frac{d}{dt} F(p(t), w(t))=G(p(t), w(t), c(t)), \end{aligned}$$

where

$$ F(p,w)=\kappa \int _\varOmega |\nabla w|^2+ \int _\varOmega p(\ln p-1) +\int _\varOmega p(w-1)- \gamma \kappa \int _\varOmega p(w-1)^2, $$

and

$$ \begin{aligned} G(p, w, c)=\,&-\displaystyle \int _\varOmega \displaystyle \frac{|\nabla p|^2}{p}+\int _\varOmega \displaystyle \frac{\alpha }{1+c}\nabla p\cdot \nabla c+ \int _\varOmega (2\alpha \gamma \kappa (1-w)+\alpha \rho ) \frac{p}{1+c}\nabla c\cdot \nabla w\\&+\displaystyle \int _\varOmega (\rho ^2-2\gamma \kappa +2\rho \gamma \kappa (1-w)) p |\nabla w|^2+ \lambda \int _\varOmega p(1-p)\ln p\\&+\displaystyle \lambda \rho \int _\varOmega p(1-p)(w-1) + \gamma \rho \int _\varOmega p^2(1-w)\\&+\displaystyle 2\gamma ^2 \kappa \int _\varOmega p^2(w-1)^2- \lambda \gamma \kappa \int _\varOmega p(1-p)(w-1)^2. \end{aligned} $$

Lemma 3.33

If $w_0>1-\displaystyle \frac{1}{\rho }$, then there exists $\kappa >0$ such that

$$\begin{aligned} G(p, w, c)\le -\frac{1}{2} \displaystyle \int _\varOmega \displaystyle \frac{|\nabla p|^2}{p}- \frac{1}{2} \displaystyle \int _\varOmega p|\nabla w|^2 +C\int _\varOmega p|w-1|+C\int _{\varOmega } p|\nabla c|^2 \end{aligned}$$

(3.5.19)

for some $C>0$.

Proof

By the Hölder and Young inequalities, we have

$$ \int _\varOmega \displaystyle \frac{\alpha }{1+c}\nabla p\cdot \nabla c\le \frac{1}{2} \displaystyle \int _\varOmega \displaystyle \frac{|\nabla p|^2}{p} +\frac{\alpha ^2}{2} \int _{\varOmega } p|\nabla c|^2, $$

and

$$ \int _\varOmega (2\alpha \gamma \kappa (1-w)+\alpha \rho ) \frac{p}{1+c}\nabla c\cdot \nabla w \le \frac{1}{2} \displaystyle \int _\varOmega p|\nabla w|^2 +C_1\int _{\varOmega } p|\nabla c|^2 $$

for some $C_1>0$.

As $w_0>1-\displaystyle \frac{1}{\rho }$, we can find some $\varepsilon _1>0$ such that $\rho (1-w_0)_+\le 1-\varepsilon _1$, where $(1-w_0)_+=\max \{0,1-w_0\}$. Hence from the w-equation in (3.1.4), it follows that

$$\begin{aligned} 1-w=(1-w_0)e^{-\gamma \int ^t_0 p(s)ds}, \end{aligned}$$

(3.5.20)

and thus

$$\begin{aligned}&\int _\varOmega (\rho ^2-2\gamma \kappa +2\rho \gamma \kappa (1-w)) p |\nabla w|^2 \\ \le&\int _\varOmega (\rho ^2-2\gamma \kappa +2\rho \gamma \kappa (1-w_0)_+) p |\nabla w|^2 \\ \le&\int _\varOmega (\rho ^2-2\gamma \kappa \varepsilon _1) p |\nabla w|^2 \\ \le&-\int _\varOmega p |\nabla w|^2 \end{aligned} $$

if we pick $\kappa >0$ sufficiently large such that $\rho ^2-2\gamma \kappa \varepsilon _1<-1$.

Denote the lower order terms of G(p, w, c) by $\theta (p, w)$, i.e.,

$$ \begin{aligned} \theta (p, w)=:&\lambda \displaystyle \int _\varOmega p(1-p)\ln p+\displaystyle \lambda \rho \int _\varOmega p(1-p)(w-1) + \gamma \rho \int _\varOmega p^2(1-w)\\&+\displaystyle 2\gamma ^2 \kappa \int _\varOmega p^2(w-1)^2- \lambda \gamma \kappa \int _\varOmega p(1-p)(w-1)^2. \end{aligned} $$

Since $s(1 - s) \ln s \le 0$ for $s\ge 0$, we get

$$ \begin{aligned} \theta (p, w) \le&\displaystyle \lambda \rho \int _\varOmega p(1-p)(w-1) + \gamma \rho \int _\varOmega p^2(1-w)\\&+\displaystyle 2\gamma ^2 \kappa \int _\varOmega p^2(w-1)^2- \lambda \gamma \kappa \int _\varOmega p(1-p)(w-1)^2\\ \le&C_2(\Vert p\Vert _{L^\infty (\varOmega )},\Vert w-1\Vert _{L^\infty (\varOmega )})\displaystyle \int _\varOmega p|w-1|,\\ \end{aligned} $$

which, along with $\Vert p(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le C$ from Theorem 3.4 and $\Vert w(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}\le \Vert w_0-1\Vert _{L^\infty (\varOmega )}$ from (3.5.20), yields

$$ \theta (p, w)\le C_3 \displaystyle \int _\varOmega p|w-1|. $$

The desired result (3.5.19) then immediately follows.

Lemma 3.34

If $w_0>1-\displaystyle \frac{1}{\rho }$, then

$$\begin{aligned} \displaystyle \sup _{t\ge 0}\int _\varOmega |\nabla w(t)|^2+ \displaystyle \int _0^\infty \int _\varOmega \displaystyle \frac{|\nabla p|^2}{p}+\displaystyle \int ^\infty _0\int _\varOmega p|\nabla w|^2<\infty . \end{aligned}$$

(3.5.21)

.

Proof

Combining Lemmas 3.30 and 3.31, we have

$$\begin{aligned} \displaystyle \frac{d}{dt} F(p(t), w(t))+\frac{1}{2} \displaystyle \int _\varOmega \displaystyle \frac{|\nabla p|^2}{p}+ \frac{1}{2} \displaystyle \int _\varOmega p|\nabla w|^2\le C\int _\varOmega p|w-1|+C\int _{\varOmega } p|\nabla c|^2. \end{aligned}$$

Hence, (3.5.21) follows upon integration on the time variable, and using (3.5.15) and (3.5.16).

Lemma 3.35

If $w_0>1-\displaystyle \frac{1}{\rho }$, then for any $r\ge 2$

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert p(\cdot ,t)-\overline{p}(t)\Vert _{L^r(\varOmega )}=0, \end{aligned}$$

(3.5.22)

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }|\overline{p}(t)-1|=0, \end{aligned}$$

(3.5.23)

where $\overline{p}(t)=\frac{1}{|\varOmega |}\int _\varOmega p(\cdot , t)$, and

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert w(\cdot ,t)-1\Vert _{L^r(\varOmega )}=0. \end{aligned}$$

(3.5.24)

Proof

The proofs of (3.5.22) and (3.5.23) are similar to those of Lemmas 5.9–5.11 of Morales-Rodrigo and Tello (2014), respectively. However, for the reader’s convenience, we only give a brief sketch of (3.5.23). In fact, from (3.1.4) and the Poincaré–Wirtinger inequality, it follows that

$$\begin{aligned} \begin{aligned} \overline{p}_t=&\lambda (\displaystyle \overline{p} - \overline{p}^2- \displaystyle \frac{1}{|\varOmega |}\int _\varOmega (p-\overline{p})^2\\ \ge&\lambda \overline{p}(\displaystyle 1-\overline{p}-C_1\int _\varOmega \frac{|\nabla p|^2}{p}). \end{aligned} \end{aligned}$$

Hence by (3.5.21), we get

$$\begin{aligned} \begin{aligned} \overline{p}(t)\ge&\displaystyle \overline{p}_0 \exp \{\lambda t -\lambda \int ^t_0 \overline{p}(s)ds-C_1\lambda \int ^\infty _0\int _\varOmega \frac{|\nabla p|^2}{p} ds \} \\ \ge&\displaystyle C_2 \exp \{\lambda t -\lambda \int ^t_0 \overline{p}(s)ds \}, \end{aligned} \end{aligned}$$

which means that (3.5.23) is valid due to either $\overline{p}(t)\rightarrow 1$ or $\overline{p}(t)\rightarrow 0$ in Lemma 5.10 of Morales-Rodrigo and Tello (2014). Indeed, suppose that $\overline{p}(t)\rightarrow 0$, then there exists $t_0>1$ such that $\overline{p}(t)\le \frac{1}{2}$ and thus $\int ^t_0 \overline{p}(s)ds \le \frac{t}{2}+ \int ^{t_0}_0 \overline{p}(s)ds$ for all $t\ge t_0$. Therefore, we arrive at $\overline{p}(t)\ge C_3 e^\frac{\lambda t}{2}$ for all $t\ge t_0$, which contradicts $\overline{p}(t)\rightarrow 0$.

Now we turn to show (3.5.24). Invoking the Poincaré inequality in the form

$$\displaystyle \int _\varOmega |\varphi (x)-\frac{1}{|\varOmega |}\int _\varOmega \varphi (y)dy|^2 dx \le C_p \int _\varOmega | \nabla \varphi |^2 dx~~\hbox { for all~} \varphi \in W^{1,2}(\varOmega )$$

for some $C_p > 0$, one can find that for all $j\in \mathbb {N}$

$$\begin{aligned} \begin{aligned} \displaystyle \int ^{j+1}_j \Vert p(s)- \overline{p}(s)\Vert ^2_{L^2(\varOmega )}ds&\le C_p\displaystyle \int ^{j+1}_j \Vert \nabla p(s)\Vert ^2_{L^2(\varOmega )}ds\\&\le C_p\displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )} \displaystyle \int ^{j+1}_j \int _\varOmega \frac{|\nabla p(s)|^2}{p(s)} ds, \\ \end{aligned} \end{aligned}$$

(3.5.25)

which, along with (3.5.21) and Theorem 3.4, shows that

$$\begin{aligned} \int _\varOmega \int ^{j+1}_j |p(x,s)- \overline{p}(s)|^2 dsdx =\int ^{j+1}_j \Vert p(s)- \overline{p}(s)\Vert ^2_{L^2(\varOmega )}ds\rightarrow 0 \end{aligned}$$

(3.5.26)

as $j\rightarrow \infty$.

Now defining $p_j(x):=\int ^{j+1}_j |p(x,s)- \overline{p}(s)|^2 ds$, $x\in \varOmega , j\in \mathbb {N}$, (3.5.26) tells us that $p_j \rightarrow 0$ in $L^1(\varOmega )$ as $j\rightarrow \infty$. There exist a certain null set $Q\subseteq \varOmega$ and a subsequence $(j_k)_{k\in \mathbb {N}}\subset \mathbb {N}$ such that $j_k\rightarrow \infty$ and $p_{j_k}(x) \rightarrow 0$ for every $x\in \varOmega \setminus Q$ as $k\rightarrow \infty$. Restated in the original variable, this becomes

$$\begin{aligned} \int ^{j_k+1}_{j_k} |p(x,s)- \overline{p}(s)|^2 ds \rightarrow 0 \end{aligned}$$

(3.5.27)

for every $x\in \varOmega \setminus Q$ as $k\rightarrow \infty$.

Therefore, from (3.5.20) and $p(x,t)\ge 0$, it follows that for any $x\in \varOmega \setminus Q$

$$\begin{aligned}&|w(x,t)-1|\nonumber \\ \le&\Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{-\gamma \displaystyle \int ^{[t]}_0 p(x,s)ds\}\nonumber \\ \le&\Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{-\gamma \displaystyle \sum _{k=0}^{m(t)}\int ^{j_k+1}_{j_k}p(x,s)ds\}\nonumber \\ \le&\Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{\gamma \displaystyle \sum _{k=0}^{m(t)}\int ^{j_k+1}_{j_k}|p(x,s)- \overline{p}(s)|ds- \gamma \sum _{k=0}^{m(t)}\int ^{j_k+1}_{j_k} \overline{p}(s)ds\}\nonumber \\ \le&\Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{\gamma \displaystyle \sum _{k=0}^{m(t)}(\int ^{j_k+1}_{j_k}|p(x,s)- \overline{p}(s)|^2 ds)^{\frac{1}{2}} -\gamma \displaystyle \sum _{k=0}^{m(t)}\int ^{j_k+1}_{j_k} \overline{p}(s)ds\},\nonumber \end{aligned}$$

where $m(t):=\displaystyle \max _{k\in \mathbb {N}}\{j_k+1,[t]\}$. Furthermore, by (3.5.23), there exists $k_0\in \mathbb {N}$ such that $\int ^{j_k+1}_{j_k} \overline{p}(s)ds\ge \frac{1}{2}$ for all $k\ge k_0$. Hence by the fact that $m(t)\rightarrow \infty$ as $t\rightarrow \infty$ and (3.5.27), we obtain that $w(x,t)-1\rightarrow 0$ almost everywhere in $\varOmega$ as $t\rightarrow \infty$. On the other hand, as $|w(x,t)-1|\le \Vert w_0-1\Vert _{L^\infty (\varOmega )}$, the dominated convergence theorem ensures that (3.5.24) holds for any $r\in (2,\infty )$.

Remark 3.4

(1) It is observed that since $W^{1,2}(\varOmega )\hookrightarrow L^\infty (\varOmega )$ is invalid in the two-dimensional setting, $\Vert p(s)- \overline{p}(s)\Vert ^2_{L^2(\varOmega )}$ in (3.5.25) cannot be replaced by $\Vert p(s)- \overline{p}(s)\Vert ^2_{L^\infty (\varOmega )}$, and thus we cannot infer that $\displaystyle \lim _{t\rightarrow \infty }\Vert w(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}=0$, even though we have established that all the related estimates of (p, c, w) in Morales-Rodrigo and Tello (2014) continue to hold under the milder condition imposed on the initial data $w_0$.

(2) Similar to the remark above, we note that, even though $\Vert w(\cdot ,t)\Vert _{W^{1,n}(\varOmega )}\le C(T)$ for any $n\ge 2$ and $t\le T$, we are not able to infer the global estimate $\sup _{t\ge 0}\int _\varOmega |\nabla w(t)|^{2+\varepsilon }$ $\le C$. Otherwise, we would be able to apply regularity estimates for bounded solutions of semilinear parabolic equations (see Porzio and Vespri (1993) for instance) to obtain the Hölder estimates of p(x, t) in $\varOmega \times (1,\infty )$, and thereby conclude $\displaystyle \lim _{t\rightarrow \infty }\Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}=0$. As things stand at the moment, we are only able to infer convergence in $L^r$.

2. $L^\infty$ -convergence of solutions with exponential rate in one dimension

It is observed that the results, in particular Lemma 3.35, in the previous subsection are still valid in the one-dimensional case. Moreover, in the one-dimensional setting, the weak convergence result in Lemma 3.35 can be improved via a bootstrap argument. In fact, we shall derive some a priori estimates of (p, c, w) and thereby demonstrate that (p, c, w) converges to (1, 0, 1) in $L^\infty (\varOmega )$ as $t\rightarrow \infty$. Furthermore, by a regularity argument involving the variation-of-constants formula for p and smoothing $L^p-L^q$ type estimates for the Neumann heat semigroup, we will show that $p(\cdot ,t)-1$ decays exponentially in $L^\infty (\varOmega )$.

As pointed out in the introduction, the main technical difficulty in the derivation of Theorem 3.5 stems from the coupling between p and w. Indeed, the lack of the regularization effect in the space variable in the w-equation and the presence of p there demand tedious estimates of the solution.

The following lemma plays a crucial role in establishing the uniform convergence of p as $t\rightarrow \infty$ (see Lemma 3.40). Though the proof thereof only involves elementary analysis, we give full proof here for the sake of the reader’s convenience since we could not find a precise reference covering our situation.

Lemma 3.36

Let k(t) be a function satisfying

$$ k(t)\ge 0, \quad \int ^\infty _0k(t)dt<\infty . $$

If $k'(t)\le h(t)$ for some $h(t)\in L^1(0,\infty )$, then $k(t)\rightarrow 0$ as $t\rightarrow \infty$.

Proof

Supposing the contrary, then we can find $A>0$ and a sequence $(t_j)_{j\in \mathbb {N}}\subset (1,\infty )$ such that $t_j\ge t_{j-1}+2$, $t_j \rightarrow \infty$ as $j\rightarrow \infty$ and $k(t_j)\ge A$ for all $j\in \mathbb {N}$. On the other hand, by $k'(t)\le h(t)$, we have

$$\begin{aligned} k(t_j-\tau )\ge k(t_j)-\int ^{t_j}_{t_j-\tau }|h(s)|ds\ge k(t_j)-\int ^{t_j}_{t_j-1}|h(s)|ds \end{aligned}$$

(3.5.28)

for all $\tau \in (0,1)$.

Since $h(t)\in L^1(0,\infty )$, we have $\int ^{t_j}_{t_j-1}|h(s)|ds\rightarrow 0$ as $t_j \rightarrow \infty$ and thereby there exists $j_0\in \mathbb {N}$ such that $\int ^{t_j}_{t_j-1}|h(s)|ds \le \frac{A}{2}$ for all $j\ge j_0$, which along with (3.5.28) implies that

$$\begin{aligned} k(t_j-\tau )\ge k(t_j)-\frac{A}{2} \ge \frac{A}{2} \end{aligned}$$

(3.5.29)

for $j\ge j_0$ and $\tau \in (0,1)$. It follows that $\int ^{t_j}_{t_j-1}k(t)dt\ge \frac{A}{2}$ for all $j\ge j_0$, which contradicts $\int ^{\infty }_{0}k(t)dt< \infty$ and thus completes the proof of the lemma.

Lemma 3.37

If $w_0>1-\displaystyle \frac{1}{\rho }$, then there exists a constant $C>0$ such that

$$\begin{aligned} \displaystyle \int _0^\infty \int _\varOmega \displaystyle e^{\rho w} {| z_x|^2}\le C \end{aligned}$$

(3.5.30)

where $z=p e^{-\rho w}$.

Proof

We know that $z_x=e^{-\rho w} p_x-\rho z w_x$ and thus

$$ e^{\rho w}|z_x|^2\le 4e^{-\rho w} |p_x|^2+4p^2 e^{-\rho w} \rho ^2 | w_x|^2. $$

Integrating over $\varOmega \times (0,\infty )$ and taking (3.5.5) into account, we have

$$ \begin{aligned} \displaystyle \int _0^\infty \int _\varOmega \displaystyle e^{\rho w} {| z_x|^2}&\le 4\displaystyle \int _0^\infty \int _\varOmega \displaystyle | p_x|^2+4\rho ^2 \displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )} \displaystyle \int _0^\infty \int _\varOmega p | w_x|^2\\&\le 4(1+ \rho ^2)\displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )} ( \displaystyle \int _0^\infty \int _\varOmega \displaystyle \frac{| p_x|^2}{p}+\displaystyle \int ^\infty _0\int _\varOmega p| w_x|^2 ). \end{aligned} $$

Hence by Theorem 3.4 and Lemma 3.34, we get (3.5.30).

Lemma 3.38

If $w_0>1-\displaystyle \frac{1}{\rho }$, then there exists a constant $C>0$ such that

$$\begin{aligned}&\displaystyle \frac{d}{dt} \int _\varOmega \displaystyle e^{\rho w} {| z_x|^2}+ \frac{1}{3}\displaystyle \int _\varOmega e^{\rho w}z_t^2 \\ \le&C ( \displaystyle \int _\varOmega e^{\rho w} {| z_x|^2}+\int _\varOmega p| w_x|^2+\int _\varOmega |c_{xx}|^2+\int _\varOmega p| c_x|^2+ \int _\varOmega p|w-1|+ \int _\varOmega p(p-1)^2 )\nonumber \end{aligned}$$

with $z=p e^{-\rho w}$.

Proof

Note that z satisfies

$$ z_t = e^{-\rho w}(e^{\rho w } z_x)_x -e^{-\rho w} (\displaystyle \frac{ z_x e^{\rho w}}{1+c} \nabla c)_x + \lambda z (1-ze^{\rho w})-\rho \gamma e^{\rho w}z^2(1-w). $$

Multiplying the above equation by $z_t e^{\rho w}$ and integrating in the spatial variable, we obtain

$$\begin{aligned} \begin{aligned}&\displaystyle \int _\varOmega e^{\rho w}z_t^2+\int _\varOmega e^{\rho w } z_x z_{xt} \\ =\,&- \displaystyle \int _\varOmega e^{\rho w}z_t(\displaystyle \frac{\alpha }{1+c} z_x c_x + \displaystyle \frac{\alpha z \rho }{1+c} w_x c_x- \displaystyle \frac{\alpha z }{(1+c)^2} | c_x|^2 +\displaystyle \frac{\alpha z }{1+c} c_{xx})\\&+\displaystyle \int _\varOmega e^{\rho w}z_t(\lambda z (1-ze^{\rho w})-\rho \gamma e^{\rho w}z^2(1-w)). \end{aligned} \end{aligned}$$

(3.5.31)

Notice that

$$ \begin{aligned} \displaystyle \int _\varOmega e^{\rho w } z_x z_{xt}=\,&\displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega \displaystyle e^{\rho w} {| z_x|^2} - \displaystyle \frac{\gamma \rho }{2} \int _\varOmega e^{2\rho w}z(1-w)| z_x|^2\\ \ge&\displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega \displaystyle e^{\rho w} {| z_x|^2} -\displaystyle \frac{\gamma \rho }{2}\displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )} \Vert 1-w_0\Vert _{L^\infty (\varOmega )}\displaystyle \int _\varOmega \displaystyle e^{\rho w} {| z_x|^2}, \end{aligned} $$

$$\begin{aligned} - \int _\varOmega z_t\frac{\alpha e^{\rho w}}{1+c} z_x c_x \ \le&\ \frac{1}{6} \int _\varOmega e^{\rho w}z_t^2 +C_1\sup _{t\ge 0}\Vert c_x (t)\Vert ^2_{L^\infty (\varOmega )} \displaystyle \int _\varOmega e^{\rho w}| z_x |^2, \\ \displaystyle \int _\varOmega z_t\displaystyle \frac{\alpha z e^{\rho w}\rho }{1+c} w_x c_x \ \le&\ \frac{1}{6}\displaystyle \int _\varOmega e^{\rho w}z_t^2 +C_1\displaystyle \sup _{t\ge 0}\Vert c_x (t)\Vert ^2_{L^\infty (\varOmega )} \displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )}\displaystyle \int _\varOmega p| w_x |^2, \\ \displaystyle \int _\varOmega \displaystyle z_t \displaystyle \frac{\alpha z e^{\rho w}}{(1+c)^2} | c_x |^2 \le&\frac{1}{6}\displaystyle \int _\varOmega e^{\rho w}z_t^2 +C_1\displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )}\displaystyle \sup _{t\ge 0}\Vert c_x (t)\Vert ^2_{L^\infty (\varOmega )} \displaystyle \int _\varOmega p| c_x |^2, \\ -\displaystyle \int _\varOmega \displaystyle z_t \displaystyle \frac{\alpha ze^{\rho w} }{1+c} c_{xx} \le&\frac{1}{6}\displaystyle \int _\varOmega e^{\rho w}z_t^2 +C_1\displaystyle \sup _{t\ge 0}\Vert p(t)\Vert ^2_{L^\infty (\varOmega )}\int _\varOmega | c_{xx} |^2 \end{aligned}$$

and

$$\begin{aligned}&\int _\varOmega e^{\rho w}z_t(\lambda z (1-ze^{\rho w})-\rho \gamma e^{\rho w}z^2(1-w)) \\ =\,&\lambda \displaystyle \int _\varOmega pz_t (1-p)-\rho \gamma \int _\varOmega \displaystyle z_tp^2(1-w)) \\ \le&\displaystyle \frac{1}{6}\displaystyle \int _\varOmega e^{\rho w}z_t^2 +C_1 \displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )}\int _\varOmega p(1-p)^2\\&+C_1 \displaystyle \sup _{t\ge 0}\Vert p(t)\Vert ^3_{L^\infty (\varOmega )} \Vert 1-w_0\Vert _{L^\infty (\varOmega )} \int _\varOmega p|1-w|. \end{aligned}$$

Applying Theorem 3.4, (3.5.18) and inserting the above inequalities into (3.5.31), we obtain the desired inequality.

Now we focus our attention on the decay properties of the solutions. Indeed, we will show that p(x, t) converges to 1 with respect to the norm in $L^\infty (\varOmega )$ as $t\rightarrow \infty$. Subsequently, we will establish the exponential decay of $\Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}$ with explicit rate.

Lemma 3.39

If $w_0>1-\displaystyle \frac{1}{\rho }$, then

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert w(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}=0. \end{aligned}$$

(3.5.32)

Proof

From (3.5.20), it follows that for any $\epsilon >0$

$$\begin{aligned} |w(x,t)-1|&\le \Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{-\gamma \int ^t_0 p(s)ds\}\\&\le \Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{\gamma \int ^t_0\Vert p(s)- \overline{p}(s)\Vert _{L^\infty (\varOmega )}ds-\gamma \int ^t_0 \overline{p}(s)ds\}\nonumber \\&\le \Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{\frac{\gamma }{\epsilon }\int ^t_0 \Vert p(s)- \overline{p}(s)\Vert ^2_{L^\infty (\varOmega )}ds+\epsilon \gamma t-\gamma \int ^t_0 \overline{p}(s)ds\},\nonumber \end{aligned}$$

(3.5.33)

where $\overline{p}(t)=\frac{1}{|\varOmega |}\int _\varOmega p(\cdot , t)$.

On the other hand, by the Poincaré–Wirtinger inequality, the Sobolev embedding theorem in one dimension and (3.5.21), we have

$$\begin{aligned} \begin{aligned} \displaystyle \int ^t_0 \Vert p(s)- \overline{p}(s)\Vert ^2_{L^\infty (\varOmega )}ds&\le C_1\displaystyle \int ^\infty _0 \Vert p_x (s)\Vert ^2_{L^2(\varOmega )}ds\\&\le C_1\displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )} \displaystyle \int ^\infty _0 \int _\varOmega \frac{| p_x (s)|^2}{p(s)} ds\\&\le C_2 \end{aligned} \end{aligned}$$

(3.5.34)

for some constant $C_2>0$. Combining (3.5.33) with (3.5.34) yields

$$\begin{aligned} \Vert w(t)-1\Vert _{L^\infty (\varOmega )}\le \Vert w_0-1\Vert _{L^\infty (\varOmega )}\exp \{\frac{C_2 \gamma }{\epsilon }+\epsilon \gamma t-\gamma \int ^t_0 \overline{p}(s)ds\} \end{aligned}$$

for $t\ge 0$. The assertion now follows from the last inequality and the proof is complete.

Lemma 3.40

If $w_0>1-\displaystyle \frac{1}{\rho }$, then

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty } \Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}=0. \end{aligned}$$

(3.5.35)

Proof

We first show that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }\Vert z(\cdot ,t)-\overline{z}(t)\Vert _{L^\infty (\varOmega )}=0 \end{aligned}$$

(3.5.36)

where $\overline{z}(t)=\frac{1}{|\varOmega |}\int _\varOmega z(\cdot , t)$.

To this end, we consider the function $k(t)\ge 0$ defined by $k(t)=\int _\varOmega \displaystyle e^{\rho w} {| z_x |^2}$ and prove that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }k(t)=0. \end{aligned}$$

(3.5.37)

By Lemmas 3.36, 3.37 and 3.38, it is enough to prove that

$$\begin{aligned} h(t)&:=\displaystyle \int _\varOmega e^{\rho w} {| z_x |^2}+\int _\varOmega p| w_x |^2+\int _\varOmega (|c_{xx}|^2+ p| c_x |^2)+ \int _\varOmega p|w-1|+ \int _\varOmega p(p-1)^2 \\&\in L^1(0,\infty ). \end{aligned}$$

Noting (3.5.30), (3.5.21), (3.5.16), (3.5.15) and (3.5.4), it remains to estimate

$$\int ^\infty _0\int _\varOmega p(p-1)^2 .$$

In fact, multiplying the p-equation in (3.1.4) by $p- 1$, we have

$$\begin{aligned} \begin{aligned} \displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega \displaystyle (p-1)^2=&- \displaystyle \int _\varOmega | p_x |^2+\rho \int _\varOmega p w_x p_x +\alpha \int _\varOmega \displaystyle \frac{p}{1+c}c_x p_x -\lambda \int _\varOmega p(p-1)^2\\ \le&-\displaystyle \frac{1}{2} \displaystyle \int _\varOmega | p_x |^2+C ( \int _\varOmega p^2| w_x |^2+ \int _\varOmega p^2| c_x |^2) -\lambda \int _\varOmega p(p-1)^2. \end{aligned} \end{aligned}$$

Hence, by the boundedness of p, (3.5.16) and (3.5.21), we easily infer that

$$ \int ^\infty _0\int _\varOmega p(p-1)^2\le C . $$

Furthermore, by the Poincaré–Wirtinger inequality and the Sobolev embedding theorem in one dimension, we have

$$\begin{aligned} \displaystyle \Vert z(t)- \overline{z}(t)\Vert _{L^\infty (\varOmega )} \le C_p\displaystyle \Vert z_x (t)\Vert _{L^2(\varOmega )}, \end{aligned}$$

which along with (3.5.37) yields (3.5.36).

On the other hand, for any $\{t_j\}_{j\in \mathbb {N}}\subset (1,\infty )$, there exists a subsequence along which $z(\cdot ,t_j)-e^{-\rho }\rightarrow 0$ a.e. in $\varOmega$ as $j\rightarrow \infty$ by Lemma 3.35. We apply the dominated convergence theorem along with the uniform majorization $|z(\cdot ,t_j)|\le \displaystyle \sup _{j \ge 1}\Vert z(t_j)\Vert _{L^\infty (\varOmega )}$ $\le C$ to infer that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty }|\overline{z}(t)-e^{-\rho }|=0. \end{aligned}$$

(3.5.38)

Hence

$$ \begin{aligned}&\Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )}\\ =&\Vert e^{\rho w}z-1\Vert _{L^\infty (\varOmega )}\\ \le&e^{\rho (1+\Vert w_0\Vert _{\infty (\varOmega )})}(\Vert z(\cdot ,t)-\overline{z}(t)\Vert _{L^\infty (\varOmega )}+|\overline{z}(t)-e^{-\rho }|)+ \Vert e^{\rho w(\cdot ,t)}-e^{\rho }\Vert _{L^\infty (\varOmega )}\\ \le&e^{\rho (1+\Vert w_0\Vert _{\infty (\varOmega )})}(\Vert z(\cdot ,t)-\overline{z}(t)\Vert _{L^\infty (\varOmega )}+|\overline{z}(t)-e^{-\rho }|)+ C_1\Vert w(\cdot ,t)-1\Vert _{L^\infty (\varOmega )} \end{aligned} $$

for some $C_1>0$, which together with (3.5.32), (3.5.36) and (3.5.38) yields the desired result.

Having established that p(x, t) converges to 1 uniformly with respect to $x\in \varOmega$ as $t\rightarrow \infty$, we now go on to establish an explicit exponential convergence rate. Using (3.5.35), we first look into the decay of $\int _\varOmega | w_x (t)|^2$.

Lemma 3.41

Let $w_0>1-\displaystyle \frac{1}{\rho }$. Then for any $\epsilon >0$, there exists $C(\epsilon )>0$ such that

$$\begin{aligned} \displaystyle \int _\varOmega | w_x (t)|^2 \le C(\epsilon )e^{-2\gamma (1-\epsilon )t}. \end{aligned}$$

(3.5.39)

Proof

From (3.5.20), it follows that

$$ \begin{aligned} | w_x (t)|^2 \le&4 | w_{0x} |^2 e^{-2\gamma \int ^t_0 p(s)ds}+4\gamma ^2| w_0-1|^2 e^{-2\gamma \int ^t_0 p (s)ds} \left( \displaystyle \int ^t_0 | p_x(s)|ds\right) ^2\\ \le&4 |w_{0x} |^2 e^{-2\gamma \int ^t_0 p(s)ds}+4 t\gamma ^2| w_0-1|^2 e^{-2\gamma \int ^t_0 p(s)ds} \displaystyle \int ^t_0 | p_x(s)|^2 ds. \end{aligned} $$

Taking Lemma 3.40 into account, we know that for any $\epsilon >0$, there exists $t_\epsilon >1$ such that $p(x,t)>1-\epsilon$ for all $x\in \varOmega , t> t_\epsilon$. Therefore, integrating the above inequality in the space variable yields

$$ \begin{aligned}&\displaystyle \int _\varOmega | w_{x}(t)|^2 \\ \le&4 e^{-2\gamma (1-\epsilon )(t-t_\epsilon )}\displaystyle \int _\varOmega |w_{0x}|^2+4 t\gamma ^2\Vert w_0-1\Vert _{L^\infty (\varOmega )}^2 e^{-2\gamma (1-\epsilon )(t-t_\epsilon )} \displaystyle \int ^\infty _0 \displaystyle \int _\varOmega | p_{x}|^2\\ \le&C_1(\epsilon )(1+t) e^{-2\gamma (1-\epsilon )t} (\displaystyle \int _\varOmega |w_{0x}|^2 + \Vert w_{0}-1\Vert _{L^\infty (\varOmega )}^2 \displaystyle \sup _{t\ge 0}\Vert p(t)\Vert _{L^\infty (\varOmega )} \displaystyle \int ^\infty _0 \displaystyle \int _\varOmega \frac{| p_x|^2}{p}), \end{aligned} $$

for all $t>t_\epsilon$, which along with (3.5.21) implies (3.5.39).

Now we utilize the decay properties of $\int _\varOmega | c_x(t)|^2$, $\int _\varOmega |w_x(t)|^2$ and the uniform convergence of $|p(x,t)-1|$ asserted by Lemma 3.40 to establish the decay property of $\Vert p(\cdot ,t)-1\Vert _{L^2(\varOmega )}$.

Lemma 3.42

Let $w_0>1-\displaystyle \frac{1}{\rho }$. Then for any $\epsilon \in (0,\min \{1,\gamma ,\lambda \})$, there exists $C(\epsilon )>0$ such that

$$\begin{aligned} \displaystyle \Vert p(\cdot ,t)-1\Vert _{L^2(\varOmega )} \le C(\epsilon )e^{-(\min \{1,\gamma ,\lambda \}-\epsilon ) t}. \end{aligned}$$

(3.5.40)

Proof

By (3.5.35), we know that for any $\epsilon \in (0,\min \{1,\gamma ,\lambda \})$, there exists $t_\epsilon >1$ such that $p(x,t)>1-\epsilon$ for all $x\in \varOmega , t> t_\epsilon$. Hence, we multiply the p-equation in (3.1.4) by $p-1$ and integrate the result over $\varOmega$ to get

$$\begin{aligned} \begin{aligned} \displaystyle \frac{1}{2}\frac{d}{dt} \int _\varOmega \displaystyle (p-1)^2&= - \displaystyle \int _\varOmega | p_x|^2+\rho \int _\varOmega pw_x p_x +\alpha \int _\varOmega \displaystyle \frac{p}{1+c} c_x p_x -\lambda \int _\varOmega p(p-1)^2\\&\le -\displaystyle \frac{1}{2} \displaystyle \int _\varOmega | p_x|^2+C_1 ( \int _\varOmega | w_x|^2+ \int _\varOmega | c_x|^2) -\lambda (1-\epsilon )\int _\varOmega (p-1)^2 \end{aligned} \end{aligned}$$

for all $t> t_\epsilon$. Now, applying the Gronwall inequality, (3.5.18) and Lemma 3.43, we have

$$\begin{aligned}&\displaystyle \int _\varOmega \displaystyle (p(t)-1)^2\nonumber \\ \le&e^{-2\lambda (1-\epsilon )(t-t_\epsilon )}\displaystyle \int _\varOmega \displaystyle (p(t_\epsilon )-1)^2 +C_1\int ^t_{0} e^{-2\lambda (1-\epsilon )(t-s)} (\displaystyle \int _\varOmega | w_x|^2+ \int _\varOmega | c_x|^2)\nonumber \\ \le&C_2(\epsilon )e^{-2\lambda (1-\epsilon )t}+C_3(\epsilon )\displaystyle \int ^t_{0} e^{-2\lambda (1-\epsilon )(t-s)} (e^{-2\gamma (1-\epsilon )s}+ e^{-2(1-\epsilon )s})\nonumber \\ \le&C_4(\epsilon )e^{-2\min \{\lambda ,1,\gamma \}(1-\epsilon )t},\nonumber \end{aligned}$$

where $c_i(\epsilon )>0$ $(i=2,3,4)$ are independent of time t. This completes the proof.

Moving forward, on the basis of Lemma 3.42, we come to establish the exponential decay of $\displaystyle \Vert p(\cdot ,t)-\overline{p}(t)\Vert _{L^\infty (\varOmega )}$ by means of a variation-of-constants representation of p, as follows.

Lemma 3.43

Let $w_0>1-\displaystyle \frac{1}{\rho }$. Then for any $\epsilon \in (0,\min \{\lambda _1,1,\gamma ,\lambda \})$, there exists $C(\epsilon )>0$ such that

$$\begin{aligned} \displaystyle \Vert p(\cdot ,t)-\overline{p}(t)\Vert _{L^\infty (\varOmega )} \le C(\epsilon )e^{-(\min \{\lambda _1,1,\gamma ,\lambda \}-\epsilon ) t}. \end{aligned}$$

(3.5.41)

Proof

By noting that $\overline{p}_t=\lambda \overline{p(1-p)}(t)$, applying the variation-of-constants formula to the p-equation in (3.1.4) yields

$$\begin{aligned} \begin{aligned} p(\cdot ,t)-\overline{p}(t)=\,&e^{t\varDelta }(p(\cdot ,0)-\overline{p}(0))-\displaystyle \alpha \int ^t_{0} e^{(t-s)\varDelta } ( \displaystyle \frac{p}{1+c} c_x)_x\\&-\rho \displaystyle \int ^t_{0} e^{(t-s)\varDelta } ( p w_x)_x +\lambda \displaystyle \int ^t_{0} e^{(t-s)\varDelta } (p(1-p)-\overline{p(1-p)}). \end{aligned} \end{aligned}$$

Together with (3.5.17), Lemmas 3.42 and 1.1, this gives

$$\begin{aligned}&\Vert p(\cdot ,t)-\overline{p}(t)\Vert _{L^\infty (\varOmega )}\nonumber \\ \le&\Vert e^{t\varDelta }(p(\cdot ,0)-\overline{p}(0))\Vert _{L^\infty (\varOmega )}+ \alpha \displaystyle \int ^t_{0}\Vert e^{(t-s)\varDelta } ( \displaystyle \frac{p}{1+c} c_x)_x\Vert _{L^\infty (\varOmega )}\nonumber \\&+\rho \displaystyle \int ^t_{0} \Vert e^{(t-s)\varDelta } ( p w_x)_x\Vert _{L^\infty (\varOmega )} +\lambda \displaystyle \int ^t_{0} \Vert e^{(t-s)\varDelta } (p(1-p)-\overline{p(1-p)})\Vert _{L^\infty (\varOmega )}\nonumber \\ \le&c_1e^{-\lambda _1 t} \Vert p(\cdot ,0)-\overline{p}(0)\Vert _{L^\infty (\varOmega )}+ C_1\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)} \Vert w_x\Vert _{L^2(\varOmega )}\\&+ C_1\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)} \Vert c_x\Vert _{L^2(\varOmega )}\nonumber \\&+ C_1\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)} \Vert p(1-p)-\overline{p(1-p)}\Vert _{L^2(\varOmega )}\nonumber \\ \le&c_1e^{-\lambda _1 t} \Vert p(\cdot ,0)-\overline{p}(0)\Vert _{L^\infty (\varOmega )}+ C_2(\epsilon )\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)}e^{-\gamma (1-\epsilon )s}\nonumber \\&+C_2(\epsilon )\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)}e^{-(1-\epsilon )s}\nonumber \\&+ C_1\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)} \Vert p(1-p)-\overline{p(1-p)}\Vert _{L^2(\varOmega )}.\nonumber \end{aligned}$$

It is observed that

$$\begin{aligned} \Vert p(1-p)-\overline{p(1-p)}\Vert ^2_{L^2(\varOmega )}=\,&\Vert p(1-p)\Vert ^2_{L^2(\varOmega )}-|\varOmega ||\overline{p(1-p)}|^2\\ \le&\Vert p(1-p)\Vert ^2_{L^2(\varOmega )}. \end{aligned}$$

Hence from (3.5.15) and Lemma 3.42, it follows that

$$\begin{aligned}&\Vert p(\cdot ,t)-\overline{p}(t)\Vert _{L^\infty (\varOmega )}\nonumber \\ \le&c_1e^{-\lambda _1 t} \Vert p(\cdot ,0)-\overline{p}(0)\Vert _{L^\infty (\varOmega )}+ C_2(\epsilon )\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)}e^{-\gamma (1-\epsilon )s}\nonumber \\&+C_2(\epsilon )\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)}e^{-(1-\epsilon )s}\\&+ C_3(\epsilon )\displaystyle \int ^t_{0}(1+(t-s)^{-\frac{3}{4}}) e^{-\lambda _1(t-s)} e^{-\min \{1,\gamma ,\lambda \}(1-\epsilon ) s}\nonumber \\ \le&C_4(\epsilon ) e^{-\min \{\lambda _1,1,\gamma ,\lambda \}(1-\epsilon ) t},\nonumber \end{aligned}$$

which implies (3.5.41).

Lemma 3.44

Let $w_0>1-\displaystyle \frac{1}{\rho }$. Then for any $\epsilon \in (0,\min \{\lambda _1,1,\gamma ,\lambda \})$, there exists $C(\epsilon )>0$ such that

$$\begin{aligned} \displaystyle |\overline{p}(t)-1| \le C(\epsilon )e^{-(\min \{2\lambda _1,2,2\gamma ,\lambda \}-\epsilon ) t}. \end{aligned}$$

(3.5.42)

Proof

We integrate the p-equation in the spatial variable over $\varOmega$ to obtain

$$\begin{aligned} \begin{aligned} (\overline{p}-1)_t=\,&\lambda (\overline{p}-\overline{p}^2-\displaystyle \frac{1}{|\varOmega |}\int _\varOmega (p-\overline{p})^2)\\ =\,&-\lambda \overline{p}(\overline{p}-1)-\displaystyle \frac{\lambda }{|\varOmega |}\displaystyle \Vert p-\overline{p}\Vert ^2_{L^2(\varOmega )}. \end{aligned} \end{aligned}$$

(3.5.43)

By (3.5.23), there exists $t_\epsilon >0$ such that $\overline{p}(t)\ge 1-\epsilon$ for $t\ge t_\epsilon$. Hence by (3.5.41) and (3.5.43), solving the differential equation entails

$$ \begin{aligned} |\overline{p}(t)-1|\le&|\overline{p}(t_\varepsilon )-1|e^{-\lambda \int ^t_{t_\varepsilon }\overline{p}(s)ds}+ \displaystyle \frac{\lambda }{|\varOmega |}\displaystyle \int ^t_{t_\varepsilon }e^{-\lambda \int ^t_{s}\overline{p}(\sigma )d\sigma }\Vert p(s)-\overline{p}(s)\Vert ^2_{L^2(\varOmega )}\\ \le&|\overline{p}(t_\varepsilon )-1|e^{-\lambda (1-\epsilon )(t-t_\epsilon )} + \displaystyle \frac{\lambda }{|\varOmega |}\displaystyle \int ^t_{t_\varepsilon }e^{-\lambda (1-\epsilon )(t-s)} \Vert p(s)-\overline{p}(s)\Vert ^2_{L^2(\varOmega )}\\ \le&|\overline{p}(t_\varepsilon )-1|e^{-\lambda (1-\epsilon )(t-t_\epsilon )} + C_1(\epsilon )\displaystyle \int ^t_{0}e^{-\lambda (1-\epsilon )(t-s)} e^{-2\min \{\lambda _1,1,\gamma ,\lambda \}(1-\epsilon ) s} \\ \le&|\overline{p}(t_\varepsilon )-1|e^{-\lambda (1-\epsilon )(t-t_\epsilon )} + C_2(\epsilon )\displaystyle e^{-\min \{2\lambda _1,2,2\gamma ,\lambda \}(1-\epsilon ) t} \\ \le&C_3(\epsilon )\displaystyle e^{-\min \{2\lambda _1,2,2\gamma ,\lambda \}(1-\epsilon ) t} \end{aligned} $$

for $t\ge t_\epsilon$, which proves (3.5.42).

Lemma 3.45

Let $w_0>1-\displaystyle \frac{1}{\rho }$. Then for any $\epsilon \in (0,\min \{\lambda _1,1,\gamma ,\lambda \})$, there exists $C(\epsilon )>0$ such that

$$\begin{aligned} \displaystyle \Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )} \le C(\epsilon )e^{-(\min \{\lambda _1,1,\gamma ,\lambda \}-\epsilon ) t}. \end{aligned}$$

(3.5.44)

Proof

Combining above two lemmas, we have

$$\begin{aligned} \displaystyle \Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )} \le \displaystyle \Vert p(\cdot ,t)-\overline{p}(t)\Vert _{L^\infty (\varOmega )}+\displaystyle |\overline{p}(t)-1| \le C(\epsilon )e^{-(\min \{\lambda _1,1,\gamma ,\lambda \}-\epsilon ) t}. \end{aligned}$$

Proof of Theorem 3.5. (3.1.7) is a direct consequence of Lemma 3.35 in the previous subsection. As for (3.1.8)–(3.1.10), we only need to collect (3.5.19), (3.5.32) and (3.5.44).

Remark 3.5

In comparison with (3.5.44), by (3.5.21) and (3.5.37), we have

$$\sup _{t\ge 0}\int _\varOmega | w_x(t)|^2+| z_x(t)|^2\le C, $$

and thus $\sup _{t\ge 0}\Vert p(t)\Vert _{W^{1,2}(\varOmega )}\le C$. Hence, an interpolation by means of the Gagliardo–Nirenberg inequality in the one-dimensional setting provides

$$\begin{aligned} \displaystyle \Vert p(\cdot ,t)-1\Vert _{L^\infty (\varOmega )} \le \displaystyle \Vert p(\cdot ,t)\Vert ^{\frac{1}{2}}_{W^{1,2}(\varOmega )}\displaystyle \Vert p(\cdot ,t)-1\Vert ^{\frac{1}{2}}_{L^2(\varOmega )} \le C(\epsilon )e^{-(\frac{1}{2}\min \{1,\gamma ,\lambda \}-\epsilon ) t}, \end{aligned}$$

where we have used (3.5.40).