Abstract
Multi-taxis appears in society interactions and cancer treatment. Society interactions can lead to the complex dynamical behavior in biology and even in criminology ([43, 65, 190]).
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6.1 Introduction
Multi-taxis appears in society interactions and cancer treatment. Society interactions can lead to the complex dynamical behavior in biology and even in criminology (Eftimie et al. 2007; Guttal and Couzin 2010; Short et al. 2008). A particular example in this direction is mixed-species foraging flocks, such as the formation of Alaska’s shearwater flocks through attraction to kittiwake foragers (Hoffman et al. 1981). Oncolytic viruses (OV) are a kind of viruses that preferentially infect and destroy cancer cells. Oncolytic viruses can be engineered by some of the less virulent viruses in nature and be readily combined with other agents. A diverse range of viruses has been investigated as potential cancer therapeutics, such as herpesvirus, adenovirus, vaccinia virus measles virus and polio virus, and oncolytic virotherapy offers a novel promising cancer treatment modality (Breitbach and Parato 2015; Goldsmith et al. 1998; Msaouel et al. 2013).
This chapter is concerned with the multi-taxis diffusion systems modeling foraging–scrounging interplay or oncolytic virotherapy. Section 6.3 is concerned with the asymptotic behavior in a doubly tactic resource consumption model with proliferation. Toward better understanding of the effect of foraging–scrounging interplay on spatio-temporal dynamics, the authors of Tania et al. (2012) proposed the forager–scrounger system given by
with positive parameters \(d, \chi _1,\chi _2,\lambda \) and nonnegative parameters \(\mu \) and r, for the unknown population densities \(u=u(x,t)\) and \(v=v(x,t)\) of foragers and scroungers and nutrient concentration \(w=w(x,t)\), respectively. The term \(-\nabla \cdot (u\nabla w)\) accounts for the tendency of foragers moving toward the increasing resource concentration, and \(-\nabla \cdot (v\nabla u)\) models the movement of scroungers following the actively searching foragers rather the resource. Due to the sequential taxis-type cross-diffusion mechanisms in (6.1.1), the considerable extra difficulties seem to be expected when compared to the corresponding scrounger-free system
Indeed, in the prototypical case \(\mu =r=0\), the two-dimensional version of (6.1.2) exhibits a substantially stronger tendency toward spatial homogeneous equilibria, which is also valid for its 3D analog at least after some waiting times (Tao and Winkler 2012c). This result implies that any destabilization of the taxis mechanism in (6.1.2) can be suppressed by the relaxation of the diffusion process together with nutrient consumption and thereby allows for a certain entropy-like structure. The feature of (6.1.1) is the sequential taxis, that is, the nutrient-taxis mechanism from (6.1.2) coupled with forager-taxis mechanism. In this situation, the mild relaxation of foragers may not suppress the potential of destabilization driven by the forager-taxis mechanism and thus limits the accessibility of energy-like techniques from the mathematical point of view. Accordingly, to the best of our knowledge, the analytical results in the literature are available only for the low dimensions or certain generalized solutions, and thereby, the comprehensive understanding of (6.1.1) is still far from complete (Black 2020; Cao 2020; Cao and Tao 2021; Liu 2019; Liu and Zhuang 2020; Tao and Winkler 2019b; Wang and Wang 2020; Winkler 2019c). For example, Tao and Winkler (2019b) established the existence of global classical solutions to the corresponding Neumann initial-boundary value problem of (6.1.1) in the one-dimensional setting for suitably regular initial data, as well as an exponential stabilization provided that the initial masses of either u or v are suitably small. As for the higher dimensional model (6.1.1), only generalized solutions are considered in Winkler (2019c) under an explicit condition on the initial datum for w and r, and moreover, they can approach spatially homogeneous equilibria in the large time limit if r decays sufficiently fast. For more related works on smooth properties of solutions to the variants of (6.1.1), inter alia accounting for the superlinear degradation mechanisms of two populations, we refer the readers to Black (2020) and Wang and Wang (2020).
On the other hand, (6.1.2) may be viewed as a kind of the predator–prey system with prey-taxis:
where u(x, t) and w(x, t) are predator density and prey density, respectively; \(\chi _1\nabla w \) is the velocity of predators pursuing preys (i.e., prey-taxis); h(u) and g(w) represent the intra-specific interaction of predators and preys, while f(w, u) is the functional response, and its typical form in the literature is \(f(w,u)=w\) (Lotka–Volterra type) and \(\frac{1}{\lambda }\) is the biomass conversion rate from the prey loss to predator gain. In contrast to the attractive Keller–Segel model, prey-taxis in most cases of (6.1.3) tends to stabilize the predator–prey interactions and may actually lead to the lack of pattern formation, which contradicts intuitive assumptions (Chakraborty et al. 2007; Lee et al. 2008, 2009; Lewis 1994). It also has been recognized that the possibility of spatial pattern formation in (6.1.3) crucially depends on the death rate of predators, the prey growth kinetics g(w) and inter alia functional forms of functional response f(w, u) (Cai et al. 2022; Lee et al. 2009; Wang et al. 2015). In addition to the pattern formation in (6.1.3), the question of which extent the intrinsic predator–prey interaction may preclude the population overcrowding has received considerable attention (see Jin and Wang 2017; Wang and Wang 2019b; Wu et al. 2018; Xiang 2018 and references therein).
In synopsis of the above results, it is natural to consider the dynamical behavior of (6.1.1) when the proliferation of foragers and scroungers is taken into account, which thus indicates that the population proliferation essentially relies on the availability of nutrient resources. Specially, this work will be concerned with the initial-boundary value problem
in a smoothly bounded domain \(\varOmega \subset \mathbb R^N\), \(N\ge 1\), where \(\nu \) denotes the outward normal vector field on \(\partial \varOmega \).
It is worthwhile to mention that (6.1.4) can be regarded as a relative of
which describes the competition between the populations u and v feeding on a common single non-renewable resource w. The authors of Krzyżanowski et al. (2019) asserted global solvability of problem (6.1.5) within a natural weak solution concept and moreover provided an analytical evidence which indicates that under suitably small initial nutrient distributions, in the long time perspective, the motility ability of population u will turn out to be a competitive advantage irrespectively of the competitive kinetics thereof. It should be remarked that the structure of (6.1.5) is comparatively simple enough to allow for the quasi-dissipative property, which seems to be lost due to the taxis-type cross-diffusive term in the second equation of (6.1.4). Inspired by Cao and Lankeit (2016), Myowin et al. (2020) and Li et al. (2019b), we shall consider the asymptotic behavior of (6.1.4) under suitably small initial data. Our standing assumptions on the initial data herein will be that
In this setting, all of the solutions of (6.1.4) approach spatially homogeneous profiles in the large time limit when suitably regular initial data satisfy a certain small condition, which reads as follows (Li and Wang 2021a). It is remarked that in comparison with the relative results of Wang and Wang (2020), the small restriction on initial data does not involve \(v_0\) herein.
Theorem 6.1
Let \(\varOmega \subset \mathbb R^N\) (\(N\ge 1\)) be a bounded domain with smooth boundary and \(m_\infty =\frac{1}{|\varOmega |}\int _\varOmega (\lambda u_0+\lambda v_0+w_0)\). Then there exists \(\varepsilon _0>0\) such that for all \(\varepsilon <\varepsilon _0\) and
with some \(p_0\in \mathbb N\) satisfying \(p_0>1+\frac{N}{2}\), the problem (6.1.4) admits a unique nonnegative global classical solution \((u,v,w)\in (C(\overline{\varOmega }\times [0,\infty ))\cap C^{2,1}(\overline{\varOmega }\times (0,\infty )))^3\). Moreover, there exist constants \(u^*\in (0,\frac{m_\infty }{\lambda })\), \(v^*\in (0,\frac{m_\infty }{\lambda })\) and \(K_i>0\) \( (i=1,2,3)\) such that for all \(t\in (0,\infty )\), we have
where \(\alpha =\min \{\lambda _1,\mu \}\) for \(\mu >0\) and \(\alpha =\min \{\lambda _1,m_\infty \}\) for \(\mu =0\) with \(\lambda _1>0\) the first nonzero eigenvalue of \(-\varDelta \) in \(\varOmega \) under the Neumann boundary condition.
It is noted that the \(L^p-L^q\) estimate for the Neumann heat semigroup \(e^{t\varDelta }\): there exists \(C>0\) such for all \(\omega \in L^q(\varOmega )\) with \(\int _\varOmega \omega =0\),
plays an important role in the derivation of decay estimations in Cao and Lankeit (2016), Myowin et al. (2020) and Li et al. (2019b). However, for the doubly tactic model (6.1.4) with \(\mu >0\), despite its dissipative feature, a more subtle effort is required in rigorous analysis due to the invalid of the mass conservation of \(\lambda u(\cdot ,t)+ \lambda v(\cdot ,t)+w(\cdot ,t)\). Indeed, the core of our argument is to verify that the interval (0, T) on which solutions enjoy some exponential decay properties can be extended to \((0,\infty )\) in which the application of above \(L^p-L^q\) estimate seems to be necessary. To this end, a nonnegative auxiliary quantity \(z(\cdot ,t)=\mu \int _0^te^{(t-s)\varDelta } w(\cdot ,s)ds\) is introduced and accordingly allows us to apply \(L^p-L^q\) estimate in our argument since the mass of \(\lambda u(\cdot ,t)+ \lambda v(\cdot ,t)+w(\cdot ,t)+z(\cdot ,t)\) is conserved now. It should be remarked that our approach is also valid when system (6.1.4) takes into account nutrient renewal r(x, t) with a certain temporal decay.
Oncolytic virotherapy offers a novel promising cancer treatment modality and currently has some limitations in the oncolytic efficacy, which might be the result of virus clearance and the physical barriers inside tumors such as the interstitial fluid pressure, extracellular matrix (ECM) deposits and tight inter-cellular junctions. The next two sections of this chapter focus on the boundedness and asymptotic behavior for solutions to oncolytic virotherapy models involving triply haptotactic terms. To better understand the physical barriers that limit virus spread, the authors of Alzahrani et al. (2019) recently proposed the PDE-ODE system of the form
to describe the coupled dynamics of uninfected cancer cells u, OV-infected cancer cells w, ECM v and oncolytic viruses (OV) z. Herein, the underlying modeling hypotheses are that in addition to random diffusion with the respective motility coefficient \(D_u\) and \(D_w\), cancer cells can direct their movement toward regions of higher ECM densities with the haptotactic coefficient \(\xi _u, \xi _w\), respectively, and that uninfected cells, apart from proliferating logistically at rate \(\mu _u\), are converted into an infected state upon contact with virus particles, whereas infected cells die owing to lysis at a rate \(\delta _w\). It is assumed that the static ECM is degraded by both types of cancer cells, possibly remodeled with rate \(\mu _v\) in the sense of spontaneous renewal of healthy tissue. Finally, it is also supposed that besides the random motion with \(D_z\) the random motility coefficient, virus particles move up the gradient of ECM with the ECM-OV-taxis rate \(\xi _z\), increase at a rate \(\beta \) due to the release of free virus particles through infected cells and undergo decay at the rate \(\delta _z\) accounting for the natural virions’ death as well as the trapping of these virus particles into the cancer cells.
From a mathematical perspective, model (6.1.7) on the one hand involves three simultaneous haptotaxis processes, but on the other hand contains the production term \(\rho uz\) in \(w-\)equation which distinguishes (6.1.7) from the most of the previous haptotaxis (Fontelos et al. 2002; Marciniak-Czochra and Ptashnyk 2010; Liţcanu and Morales-Rodrigo 2010b; Tao 2011; Winkler 2018b) and chemotaxis–haptotaxis models (Pang and Wang 2018; Stinner et al. 2014; Tao and Winkler 2019a). In fact, the haptotactic migration of u, z toward higher densities v simultaneously, in which no smoothing action on the spatial regularity of v can be expected, renders us unable to apply smoothing estimates for the Neumann heat semigroup to gain a priori boundedness information on u and z beyond the norm in \(L^1 (\varOmega )\). Accordingly, this superlinear production term \(\rho uz\) in (6.1.7) seems likely to increase the destabilizing potential in the sense of enhancing the tendency toward blow-up of solutions and thus becomes the key contributor to mathematical challenges already given in the derivation of global solvability theory of (6.1.7), which is also indicated in the qualitative analysis of chemotaxis-May–Nowak model (Bellomo and Tao 2020; Bellomo et al. 2019; Hu and Lankeit 2018; Winkler 2019d).
Though the methodological limitations seem to widely restrict the theoretical understanding of the full model (6.1.7), some analytical works on simplifications of the latter have recently been achieved in Tao and Winkler (2020a, 2020b, 2021). Indeed, upon neglecting haptotactic migration processes of infected tumor cells and oncolytic viroses, renewal of ECM as well as proliferation of infected tumor cells, Tao and Winkler considered the corresponding Neumann initial-boundary value problem for
in a bounded domain \(\varOmega \subset \mathbb {R}^2\) and obtained that the globally defined classical solution is bounded if \(0<\beta <1\), \(\rho \ge 0\) (Tao and Winkler 2020a), whereas for \(\beta >1\) and \(\int _{\varOmega }u(\cdot ,0)>|\varOmega |/(\beta -1)\), infinite-time blow-up occurs at least in the particular case when \(\rho =0\) (Tao and Winkler 2021). In order to provide an complement to this, the study in Tao and Winkler (2022) reveals that for any \(\rho \ge 0\) and arbitrary \(\beta >0\), at each prescribed level \(\gamma \in (0,1/(\beta -1)_+)\), one can identify an \(L^\infty \)-neighborhood of the homogeneous distribution \((u,v,w,z)\equiv (\gamma ,0,0,0)\) within which all initial data lead to globally bounded solutions that stabilize toward the constant equilibrium \((u_\infty ,0,0,0)\) with some \(u_\infty >0\). On the other hand, in Tao and Winkler (2020c), it is proved that if \(\beta \in (0,1)\), for any choice of \(M>0\), one can find initial data such that the globally defined classical solution satisfies \(u\ge M\) in \(\varOmega \times (0,\infty )\).
Moreover, for the doubly haptotactic version of (6.1.7) with \(\xi _z=0\), the global classical solvability to the corresponding initial-boundary value problem for more comprehensive systems of the form
is proved in Tao and Winkler (2020b). This is achieved by discovering a quasi-Lyapunov functional structure that allows to appropriately cope with the presence of nonlinear zero-order interaction terms which apparently form the most significant additional mathematical challenge of the considered system in comparison to previously studied haptotaxis models.
The purpose of Sect. 6.4 is to a more comprehensive understanding of model (6.1.7) in the biologically most relevant constellation in which the haptotactic motion of virus particles is taken into account particularly, and either the production term uz or proliferating term \(\mu _u u(1-u)\) is adjusted to \(\displaystyle \frac{uz}{k_u + \theta u}\) of the Beddington–deAngelis type with positive parameters \(k_u,\theta \) (Bellomo and Tao 2020) or \(\mu _u u(1-u^r)\) of superquadratic type, respectively. We are concerned with the PDE-ODE system given by
in a bounded domain \(\varOmega \subset \mathbb {R}^2\) with smooth boundary, where for the initial data \((u_0,w_0,v_0, z_0)\), we suppose throughout Sect. 6.4 that
Beyond the global classical solvability, in Sect. 6.4, we focus on the global boundedness of classical solutions to (6.1.10)–(6.1.11) stated as follows, which can be regarded as a first step toward the qualitative comprehension of (6.1.10) (Li and Wang 2021b).
Theorem 6.2
Let \(\varOmega \subset \mathbb {R}^2\) be a bounded domain with smooth boundary, \(D_u,D_w\), \(D_z\), \(\xi _u,\xi _w, \xi _z,\mu _u,\mu _v,\rho ,k_u,\alpha _u,\alpha _w,\beta ,\delta _w\) and \(\delta _z \) are positive parameters. Suppose that \(r=1,\theta >0\) or \(r>1,\theta \ge 0\). Then for any choice of \((u_0, w_0, v_0, z_0)\) fulfilling (6.1.11), there exists \(C>0\) such that if \(\xi _w \alpha _w<C\), (6.1.10) admits a unique global classical solution (u, w, v, z), where \(\Vert u(\cdot ,t)\Vert _{L^\infty {(\varOmega )}}\),\(\Vert w(\cdot ,t)\Vert _{L^\infty {(\varOmega )}}\) and \(\Vert z(\cdot ,t)\Vert _{L^\infty {(\varOmega )}}\) are uniformly bounded for \(t\in (0,\infty )\).
Remark 6.1
In line with the above discussion, the boundedness result on (6.1.10) with \(r=1,\theta =0\) is also valid when \(\xi _z=0\).
Remark 6.2
When \(\mu _v=0\), one can see that the restriction on \(\xi _w \alpha _w\) in Theorem 6.2 can be replaced by a certain small condition on \(v_0\).
A cornerstone of our analysis is to show that for the suitably small \(\xi _w \alpha _w\), the functional
with \(a= ue^{-\chi _u v}, b= we^{-\chi _w v}\) and \(c= ze^{-\chi _z v}\) enjoys a certain quasi-dissipative property under appropriate choice of the positive constant A (of (6.4.32)). As the first step in this direction, we perform the variable change used in several precedents, by which the crucial haptotactic contribution to the equations in (6.1.10) is reduced to zero-order terms \(\chi _u a(\alpha _u u+\alpha _w w)v-\chi _u\mu _v av(1-v)\), \(\chi _w b(\alpha _u u+\alpha _w w)v-\chi _w\mu _v bv(1-v)\) and \( \chi _z c(\alpha _u u+\alpha _w w)v-\chi _z\mu _v cv(1-v)\), respectively (see (6.4.1) below). Thanks to a variant of the Gagliardo–Nirenberg inequality involving certain \(L\log L\)-type norms, the latter offers the sufficient regularity so as to allow for the \(L^\infty \)-bounds of solutions in the present two-dimensional setting.
Section 6.5 is devoted to understand the dynamics behavior of (6.1.7) to a considerable extent in higher dimensional settings in light of the above-mentioned results and a recent consideration of global classical solutions to the one-dimensional (6.1.7) in Tao (2021). Specially, taking into account the linear degradation instead of the renewal of ECM and neglecting the proliferation of uninfected tumor cells in (6.1.7), we are concerned with the following Neumann initial-boundary problem in \(\varOmega \subset \mathbb R^N(N\ge 1)\):
where \(\xi _u\), \(\xi _w\), \(\xi _z\), \(\rho _u\), \(\rho _w\), \(\rho _z\), \(\delta _w\), \(\delta _v\), \(\delta _z\), \(\alpha _u\), \(\alpha _w\) and \(\beta \) are positive parameters, for the initial data \((u_0,w_0,v_0, z_0)\), we suppose throughout the third part of Chap. 6 that
Our main result makes sure that for suitably small initial data, these solutions will be globally bounded and approach some constant profiles asymptotically (Wei et al. 2022).
Theorem 6.3
Let \(\varOmega \subset \mathbb {R}^{N}\left( N\ge 1 \right) \) be a bounded domain with smooth boundary. Assume (6.1.13) holds and \(\beta \le \frac{\rho _{u}+\rho _{z}}{\rho _{w}}\delta _{w}\). Then if for some \(p_0>\max \{1,\frac{N}{2}\}\), there exists \(\varepsilon >0\) which depends on \(\xi _u, \xi _w, \xi _z, \rho _u, \rho _w, \rho _z, \alpha _y, \alpha _w\) such that
the problem (6.1.12) has a unique nonnegative global classical solution
Moreover, there exists a nonnegative constant \(u^{*}\) such that
as \(t\rightarrow \infty \).
Remark 6.3
Our result indicates that the infected cancer cells and virus particle population can become extinct asymptotically and the density of uninfected cancer cells tends to a nonnegative constant \(u^*\) which is less than \(\overline{u}_0\). This result implies that the oncolytic virotherapy is effective. Unfortunately, the condition under which \(u^*\) equals to zero is left as an open problem.
Same to the analysis in Sect. 6.3, a more subtle effort seems to be required for our analysis in Sect. 6.5 due to the decreasing of \(\int _{\varOmega }\left( u+\frac{\rho _{u}+\rho _{z}}{\rho _{w}}w+z\right) (\cdot ,s)ds\). To this end, a nonnegative auxiliary quantity
is introduced and accordingly allows us to apply \(L^p-L^q\) estimate in our argument since the mass of \(u+\frac{\rho _{u}+\rho _{z}}{\rho _{w}}w+z+Q\) is conserved now.
6.2 Preliminaries
In this section, we provide some preliminary results that will be used in the subsequent sections.
By applying the maximal Sobolev regularity (Theorem 3.1 of Hieber and Prüss 1997), we can obtain the following lemma, which together with Lemmas 1.1, 3.2 and 4.3 will play an important role in the proof of our main results in Sects. 6.3 and 6.5.
Lemma 6.1
(Ishida et al. 2014, Lemma 2.1; Yang et al. 2015, Lemma 2.2) Let \(r\in (1,\infty )\) and consider the following evolution equation:
Then for each \(h_0\in W^{2,r}(\varOmega )\) with \(\nabla h_0\cdot \nu =0\) on \(\partial \varOmega \) and any \(f\in L^r((0,T),L^r(\varOmega ))\), (6.2.1) admits a unique mild solution \(h\in W^{1,r}((0,T);L^r(\varOmega ))\cap L^r((0,T);W^{2,r}(\varOmega ))\). Moreover, there exists \(C_r>0\), such that
The following lemma is a special case of Lemma A.5 in Tao and Winkler (2014b) and can be regarded as a variant of a Gagliardo–Nirenberg inequality originally derived in Biler et al. (1994), which will be of importance in the later analysis in Sect. 6.4.
Lemma 6.2
Let \(\varOmega \subset \mathbb {R}^2\) be a bounded domain with smooth boundary, and let \(p \in (1,\infty )\) and \(\varepsilon > 0\). Then there exists \(K(p,\varepsilon ) >0\) such that
holds for all \(\varphi \in W ^{1,2}(\varOmega )\).
The following Lemma is based on simple calculations on the maximal value of the function \(f(t)=t^ae^{-b t}\) for \(t\ge 0\), which is used in the analysis in Sect. 6.5.
Lemma 6.3
For all \(a>0\), \(b>0\), we have
holds for all \(t\ge 0\).
6.3 Asymptotic Behavior of Solutions to a Doubly Tactic Resource Consumption Model
At the beginning of this subsection, we provide a basic state on local existence and extensibility of solutions to (6.1.4), which can be readily proved by the Amann theory. Similar proof thereof can be found in Tao and Winkler (2019b, Lemma 2.1) and hence we omit the detail here.
Lemma 6.4
There exist a maximal existence time \(T_{max}\in (0,\infty ]\) and \((u,v,w)\in C(\overline{\varOmega }\times [0,T_{max}))\cap C^{2,1}(\overline{\varOmega }\times (0,T_{max}))\) such that (u, v, w) is the unique nonnegative solution of (6.1.4) in \([0,T_{max})\). Furthermore, if \(T_{max}<+\infty \), then
for all \(q>N\).
Theorem 6.1 is the consequence of the following lemmas. In the proof of these lemmas, the constants \(c_i>0\), \(i=0,1,\ldots ,4\), refer to those in Lemmas 1.1 and 4.3, respectively.
We first collect some easily verifiable observations in the following lemma:
Lemma 6.5
Under the assumptions of Theorem 6.1, there exist \(M_i>1(i=1,2,3),\) and \(\varepsilon >0\) such that
where
and constant \(C_5>0\) is given in Lemma 6.10 below which is independent of \(M_i(i=1,2,3)\).
To obtain a conservation law of mass, we introduce an nonnegative variable z satisfying
then it is easy to see that for any \(t\in [0,T)\),
Let
By Lemma 6.4 and the smallness condition on initial data in Theorem 6.1, \(T>0\) is well-defined. We first show \(T=T_{max}\). To this end, we will show that all of the estimates mentioned in (6.3.7) are valid with even smaller coefficients on the right-hand side. The derivation of these estimates will mainly rely on \(L^p-L^q\) estimates for the Neumann heat semigroup and the fact that the classical solutions on \((0,T_{max})\) can be represented as
for all \(t\in (0,T_{max})\) as per the variation-of-constants formula.
Lemma 6.6
Suppose that the assumptions from Theorem 6.1 hold. Then for all \(t\in (0,T)\), we have
with \(M_5:=M_1+2c_1\).
Proof
Due to
and the assumption of Theorem 6.1, the definition of T along with Lemma 1.1(i) implies that for all \(t\in (0,T)\),
Lemma 6.7
Under the assumptions of Theorem 6.1, we have
Proof
Multiplying the third equation in (6.1.4) by \(kw^{k-1}\) and integrating the result over \(\varOmega \), we get \(\frac{d}{dt}\int _\varOmega w^k\le -k\int _\varOmega (\lambda u+\lambda v+\mu ) w^k\) on (0, T).
In what follows, we shall show (6.3.14) in two cases: \(\mu >0\) and \(\mu =0\).
(I) The case \(\mu >0\). Since \(-(\lambda u+\lambda v+\mu )\le -\mu \), we have
and thus
which implies that
for any \(t\in (0,T)\), where we have used (6.3.2).
(II) The case \(\mu =0\). Note that \(z\equiv 0\) for all \((x,t)\in \varOmega \times [0,T)\), and thus
From the definition of T and Lemma 6.6, it follows that for any \(t\in (0,T)\),
and hence
which implies that for any \(t\in (0,T)\),
Thanks to \(\Vert w_0\Vert _{L^\infty (\varOmega )}\le \varepsilon \) and (6.3.2), (6.3.4), we obtain that for any \(t\in (0,T)\),
by letting \(k\rightarrow \infty \) in (6.3.15).
Lemma 6.8
Let the conditions from Theorem 6.1 be fulfilled. Then for all \(t\in (0,T)\) and \(p_0>\frac{N}{2}\), we have
with \(M_4:=2 c_3+2|\varOmega |^{\frac{1}{2p_0}}\left( 1+m_\infty +\mu \right) c_{10}c_2M_2\).
Proof
By (6.3.11) and Lemma 1.1(iii), noticing \(\Vert \nabla w_0(\cdot )\Vert _{L^{2p_0}(\varOmega )}\le \varepsilon \), we have
Now, we estimate the last two integrals on the right-hand side of the above inequality. From the definition of T, Lemmas 1.1(ii), 4.3 and 6.6, it follows that
Inserting (6.3.17) into (6.3.16), we get
and thereby complete the proof.
Lemma 6.9
Under the assumptions of Theorem 6.1, for all \(p_0>\frac{N}{2}\), there exists a constant \(C>0\) independent of T such that
Proof
Noticing that w satisfies
and u satisfies
with \(F(x,t)=-\lambda (u+v)w-\mu w\) and \(G(x,t)=-\chi _u(\nabla u\nabla w+u\varDelta w)+uw\), respectively. By Lemmas 6.6 and 6.7, we can see that
Hence, thanks to Lemma 6.1, we can find \(C_1>0\) independent of T such that
Similarly, by the definition of T, (6.3.18) and Lemma 6.8, there exists \(C_2>0\) independent of T fulfilling
Applying Lemma 6.1 to (6.3.21) once more, we have
for some \(C_3>0\) independent of T and thereby complete the proof.
Thanks to the decay property of \(w,v, \nabla u\) and space–time \(L^{p_0}\)-estimate for \(\triangle u\), we can establish an \(L^{2(p_0-1)}\) bound for \(\nabla v\) based on Lemmas 1.1 and 4.3.
Lemma 6.10
Suppose that the requirements from Theorem 6.1 are met. Then for all \(p_0>1+\frac{N}{2}\), there exists \(C_5>0\) independent of T and \(M_i(i=1,2,3)\) such that
Proof
By (6.3.10), we have
From Lemma 1.1(iii), we obtain
Now, we estimate the last two integrals on the right-hand side of (6.3.23). From the definition of T, Lemmas 1.1(ii), 4.3, 6.6 and 6.9, it follows that
where \(C_1:=\int _0^t\Vert \varDelta u(\cdot ,s)\Vert _{L^{p_0}(\varOmega )}^{p_0}ds\) is bounded by Lemma 6.9 and
for \(p_0>1+\frac{N}{2}.\)
Next, by Lemmas 1.1(ii), 4.3, 6.6 and 6.7, we get
Inserting (6.3.24)–(6.3.26) into (6.3.23) and using (6.3.4), we readily get
and thereby from Lemma 6.5, we arrive at
with some \(C_5>0\) independent of T, \(M_i(i=1,2,3)\) and hence complete the proof.
Beyond the weak information of \(\triangle w\) in Lemma 6.9, we now turn the boundedness of \(\nabla v\) into a statement on decay of \(\triangle w\).
Lemma 6.11
Under the assumptions of Theorem 6.1, for all \(p_0>\frac{N}{2}\),
with
Proof
From (6.3.8), we have
From Lemma 1.1(i) and the fact that \(\Vert \varDelta w_0\Vert _{L^{p_0}(\varOmega )}\le \varepsilon \), we obtain that
Now, we estimate the last two integrals on the right-hand side of the above inequality. From the definition of T, Lemmas 1.1(iv), 4.3, 6.6, 6.8, 6.10 and (6.3.4), it follows that
and
Inserting (6.3.30), (6.3.31) and (6.3.32) into (6.3.29), we obtain
and thereby complete the proof.
Lemma 6.12
Under the assumptions of Theorem 6.1, for all \(p_0>\frac{N}{2}\),
Proof
By (6.3.9), we have
From Lemma 1.1(iii) and the fact that \(\Vert \nabla u_0(\cdot )\Vert _{L^{2p_0}(\varOmega )}\le \varepsilon \), we obtain
Now, we estimate the last two integrals on the right-hand side of (6.3.34). From the definition of T, Lemmas 1.1(ii), 4.3, 6.6, 6.8 and 6.11, it follows that
and
Inserting (6.3.35)–(6.3.37) into (6.3.18) and using (6.3.3), we readily get
and thereby complete the proof.
Lemma 6.13
Under the assumptions of Theorem 6.1, for all \(t\in (0,T)\),
Proof
According to (6.3.7) and Lemma 1.1(iv), we have
Now, we need to estimate \(I_1\) and \(I_2\). Firstly, from the definition of T, Lemmas 4.3, 6.6 and 6.8, we obtain
and
Combining (6.3.38)–(6.3.40) along with (6.3.1) leads to
and hence ends the proof.
Now, we have prepared the major parts of the proof of Theorem 6.1 and thus can verify asymptotic properties stated there.
Proof of the Theorem 6.1. First we claim that \(T=T_{max}\). In fact, if \(T<T_{max}\), then by Lemmas 6.7, 6.12 and 6.13, we have
for all \(t\in (0,T)\), which contradicts the definition of T in (6.3.7).
Next, we show that \(T_{max}=\infty \). In fact, if \(T_{max}<\infty \), then in view of the definition of T, Lemmas 6.6, 6.8 and 6.10, we obtain that for any \(p_0>1+\frac{N}{2}\),
which contradicts with Lemma 6.4. Therefore, we have \(T_{max}=\infty \).
Integrating the first equation in (6.1.4) over \(\varOmega \), we have
which, along with the nonnegative property of u, w and the fact that
warrants
as well as
As a consequence of the latter, we immediately have
On the other hand, by Poincare’s inequality,
and thanks to \(W^{1,2p_0}(\varOmega )\hookrightarrow L^{\infty }(\varOmega )\) for \(p_0>\frac{N}{2}\), we can find \(C_2>0\) such that
Therefore, by (6.3.43) and the fact that \(\Vert \nabla u\Vert _{L^{2p_0}(\varOmega )}\le M_3\varepsilon e^{-\alpha t}\), we can pick \(K_1>0\) such that
On the other hand, from (6.3.12) and Lemma 1.1(ii), we infer that
By similar procedure as that in the derivation of (6.3.45), there exists constant \(K_2>0\) such that
with
Then from the fact that
with
using (6.3.13), (6.3.14), (6.3.45) and (6.3.47), there exists \(K_3>0\) such that
The decay estimates claimed in Theorem 6.1 readily follow and the proof of this theorem is thus completed.
6.4 Boundedness of Solutions to an Oncolytic Virotherapy Model
6.4.1 Some Basic a Prior Estimates
For the convenience in our subsequent estimation procedure, we let
and introduce the variable change used in several precedents (Fontelos et al. 2002; Pang and Wang 2018; Tao and Winkler 2014b)
upon which (6.1.10) takes the following form:
with
It is noted that (6.1.10) and (6.4.1) are equivalent in this framework of classical solutions. The following basic statement on the local existence and extensibility criterion of classical solutions to (6.4.1) can be proved by a straightforward adaptation of the reasoning in Pang and Wang (2018) and Tao and Winkler (2020b).
Lemma 6.14
Let \(D_u,D_w\), \(D_z\), \(\xi _u,\xi _w, \xi _z,\mu _u,\mu _v,\rho ,k_u,\alpha _u,\alpha _w,\beta ,\delta _w\) and \(\delta _z \) are positive parameters, and assume that \(r\ge 1,\theta \ge 0\). Then there exist \(T_{max}\in (0,\infty ]\) and a uniquely determined quadruple \((a,b,v,c)\in (C^{2,1}(\overline{\varOmega } \times [0,T_{max})))^4\) which solves (6.4.1) in the classical sense and \(a>0,b>0,c>0\) and \(v>0\) in \(\varOmega \times (0,T_{max})\), and that if \(T_{max}<+\infty \), then
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 (6.4.2) (cf. Pang and Wang 2018; Tao and Winkler 2014b for instance). With the help of the maximum principle, we can also verify the asserted positivity of the solutions.
From now on without any further explicit mentioning, we shall suppose that the assumptions of Theorem 6.2 are satisfied, and let (a, b, v, c) and \(T_{max}\in (0,\infty ]\) be as provided by Lemma 6.14. Moreover, we may tacitly switch between these variables and the quadruple (u, w, v, z) if necessary.
The following important properties of solutions of (6.1.10) can be easily checked.
Lemma 6.15
Let \(T>0\). Then solution (u, w, v, z) of (6.1.10) satisfies
and
for all \(t\in (0, \hat{T})\) as well as
where \(\hat{T}:=\min \{T, T_{max}\}\).
Proof
Integrating the first equation in (6.1.10) over \(\varOmega \) yields
due to \(z\ge 0\). Since \( (\int _{\varOmega }u)^{r+1}\le |\varOmega |^{r}\int _{\varOmega }u^{r+1}\) by the Cauchy–Schwartz inequality, (6.4.7) implies that \(y(t):=\int _{\varOmega }u(\cdot ,t)\) satisfies
from which (6.4.3) follows by the Bernoulli inequality. On the other hand, due to the nonnegativity of u, w and v in \(\overline{\varOmega }\times (0, \hat{T})\), the comparison principle entails that \(v_t\le \mu _v v(1-v)\) and thus the estimate in (6.4.4) follows similarly.
Once more integrating the equations in (6.1.10) over \(\varOmega \) and using the fact that \(2u\le u^{r+1}+4 \), we can see that
and
as well as
Combining (6.4.8)–(6.4.9), we obtain that
which entails that \(y(t):=\int _{\varOmega }u+ \int _{\varOmega } w\) satisfies
Hence, using the Bernoulli inequality to the above inequality, we get the estimate in (6.4.5). Further, it follows from (6.4.10) that
and thereby derive (6.4.6) by an ODE comparison argument.
6.4.2 Bounds for a, b and c in LlogL
This section aims to construct an Lyapunov-like functional involving the logarithmic entropy of a, b and c, rather than that of u, w and z, which provides some regularity information of solutions that forms the crucial step in establishing \(L^\infty \) bounds for u, w and z in the present spatially two-dimensional setting. It should be mentioned that upon the special structure of (6.1.9), inter alia neglecting haptotactic migration processes of oncolytic viruses z, the energy-like functional \(\mathscr {F}\) in Tao and Winkler (2020b) can be achieved by appropriately combining the logarithmic entropy of u, w, Dirichlet integral of \(\sqrt{v}\) and integral of \(z^2\) in line with some precedent studies (see Tao and Winkler 2014b; Winkler 2018b).
The first step of our approaches consists in testing the first equation of (6.4.1) against \(\ln a\).
Lemma 6.16
For any \(\varepsilon \in (0,1)\), there exists \(K_1(\varepsilon )>0\) such that
Proof
From the first equation in (6.4.1), it follows
By the positivity of a in \(\overline{\varOmega }\times (0,\infty )\), testing the first equation in (6.4.1) by \(\ln a\) then shows that
By (6.4.6), we see that
since \(a\ln a\ge -e^{-1}\) for all \(a>0\) and \(v(x,t)\le v^*\) for all \(x\in \varOmega ,t>0\) by (6.4.4). Apart from that, for any \(\varepsilon \in (0,1)\), there exists \(C_1(\varepsilon )>0\) such that
due to \(a^2\le \varepsilon _1 a^2\ln a+e^{\frac{2}{\varepsilon _1}}\), \(a\ln a\le \varepsilon _1 a^2\ln a- \varepsilon _1^{-1} \ln \varepsilon _1 \) and \(a\le \varepsilon _1 a^2\ln a+2 e^{\frac{2}{\varepsilon _1}}\) for any \(\varepsilon _1\in (0,1)\). Therefore, inserting above two inequalities into (6.4.12), we arrive at (6.4.11).
Lemma 6.17
There exists \(c^*>0\) with the property that if \(\xi _w\alpha _w<c^*\), then one can find \(\varepsilon _0\in (0,1)\) and \(K_2>0\) such that for all \(\varepsilon \in (0,\varepsilon _0)\),
where \(r=1\) if \(\theta >0\) and \(r>1\) if \(\theta \ge 0\).
Proof
From the second equation in (6.4.1), it follows that
By straightforward calculation relying on \(0\le v\le 1\) in \(\overline{\varOmega }\times (0,\infty )\) and the Young inequality, we then see that for any \(\varepsilon >0\),
The first summand on the right-hand side of (6.4.14) will be estimated in the case \(r=1, \theta >0\) and \(r>1, \theta \ge 0\), respectively.
For \(r=1\) and \(\theta > 0\), we have
Since \(\ln ^{2} s\le \frac{4}{ e^2}s \) for all \(s>1\), an application of the Hölder inequality leads to
In conjunction with (6.4.15), we can see that
To estimate the first term on the right-hand side of (6.4.16), by means of the two-dimensional Gagliardo–Nirenberg inequalities, we can find \(K_g>0\) such that
Thereby thanks to (6.4.4) and (6.4.5), there exists \(C_1>0\) such that
provided that \(\xi _w\alpha _w<c^*:=\frac{2 D_w}{e^{2\chi _w v^* }K_g w^*v^*}\) and any \(0<\varepsilon <\varepsilon _{0}:=\min \{1, \frac{D_w }{e^{2\chi _w v^* }K_g w^*v^*}\}\). Therefore, along with Lemma 6.15 and the Hölder inequality, we insert (6.4.18) into (6.4.16) to arrive at (6.4.13).
While for \(r>1, \theta \ge 0\), we have
Here, an apparently challenging issue is to estimate \( \int _\varOmega z u\ln b\) appropriately in terms of expression which can be controlled by the dissipation terms in (6.4.11). Since there exists \(C_2>0\) such that \(\ln ^\frac{2(r+1)}{r-1} s\le s+C_2\) for all \(s>1\), we can infer from (6.4.6) and the Hölder inequality that
with \(C_3=\frac{\rho ^2}{k_u^2} (\Vert b \Vert _{L^1(\varOmega )}+C_2|\varOmega |)^{\frac{r-1}{r+1}}\).
Apart from that, by the Hölder inequality and the Young inequality, it is easy to see that
In conjunction with (6.4.19), we get
for some \(C_4>0\), which together with (6.4.18) and (6.4.5) implies that (6.4.13) holds.
Lemma 6.18
There exists \(K_3>0\) such that
Proof
By the fourth equation in (6.4.1), we can see that
Proceeding as above, we test (6.4.21) by \(\ln c\) and integrate by parts to see that for \(\varepsilon >0\),
due to
and
Now according to the two-dimensional Gagliardo–Nirenberg inequality (6.4.17) and Lemma 6.15, we pick \(\varepsilon =\frac{D_z}{8K_gz^*}\) and thereafter obtain some \(C_1>0\) such that
Therefore, along with Lemma 6.15, in conjunction with (6.4.22) and (6.4.23), we readily arrive at (6.4.20).
We are now ready to obtain the bounds for a, b and c in LlogL by taking suitable linear combinations of the inequalities provided by Lemmata 6.16–6.18, stated as follows.
Lemma 6.19
Let \(T>0\). Then there exists \(K_4> 0\) such that
and
for all \(t\in (0, \hat{T})\) with \(\hat{T}:=\min \{T, T_{max}\}\).
Proof
From (6.4.18) and (6.4.23), it follows that there exists \(C_1>0\) such that
Multiplying (6.4.13) by \(A:=\frac{8K_3C_1}{D_w}\) and adding the resulting inequality to (6.4.20), using (6.4.27) and (6.4.28), we have
Taking \(\varepsilon =\frac{3D_z}{8AC_1}\) in (6.4.29), we can find \(C_2>0\) such that
Combining (6.4.11) with (6.4.30) and using (6.4.18), we can pick \(\varepsilon >0\) in (6.4.11) appropriately small to derive that for some \(C_3>0\)
which, along with \(a^{r+1}\le \varepsilon a^{r+1}\ln a+c(\varepsilon )\) for some \(c(\varepsilon )>0\), implies that there exist \(C_4>0\) and \(C_5>0\) fulfilling
with
and thereby
is valid for some \(C_6>0\).
Now, by the inequality \(a\ln a\ge -e^{-1}\) for all \(a>0\),
and similarly,
as well as
Hence, (6.4.24)–(6.4.26) result readily from (6.4.33).
6.4.3 \(L^\infty \)-Bounds for a, b and c
By means of some quite straightforward \(L^p\) testing procedures, combining Lemma 2.1 with appropriate interpolation, we can now proceed to turn the outcome of Lemma 6.19 into the \(L^\infty \)-bounds for a, b and c.
Lemma 6.20
Let (a, b, v, c) be the classical solution of (6.4.1) in \(\varOmega \times [0,T_{max})\). Then one can find \(C>0\) fulfilling
and
as well as
for all \(t\in (0,T_{max})\).
Proof
Testing the first equation in (6.4.1) by \( e^{\xi _u v} a^{p-1}\) with \(p>4\), integrating by parts and using the Young inequality, we can find \( C_1>0\) and \(C_2:=C_2(p)>0\) such that
Similarly, based on the other equations in (6.4.1), we infer the existence of \(C_3 > 0\) such that
as well as
Collecting (6.4.37)–(6.4.39), we then have
Now on the basis of Lemma 6.19, we employ Lemma 2.1 to estimate \( \int _\varOmega a^{p+1}\), \( \int _\varOmega b^{p+1}\) and \(\int _\varOmega c^{p+1}\) in term of \(\int _\varOmega |\nabla a^{\frac{p}{2} }|^2\), \(\int _\varOmega |\nabla b^{\frac{p}{2} }|^2\) and \(\int _\varOmega |\nabla c^{\frac{p}{2} }|^2\), respectively.
Indeed, applying Lemma 2.1 to \(\varphi =a^{\frac{p}{2}}\), we have
which along with (6.4.24) and the appropriate choice of \(\varepsilon \) readily shows that for \(C_4(p)>0\)
Similarly,
as well as
Therefore, (6.4.40) shows that
which entails that for all \(p\ge 2\) there exists \(C_8(p) > 0\) such that
for all \(t\in (0,T_{max})\).
Furthermore, by adapting a well-established Moser-type iteration, one can readily turn the latter into the \(L^\infty \) bounds for a, b, c. However, since the procedure is rather standard (see Tao and Winkler 2014b, 2020b for example), we give the details only in places which are characteristic of the present setting.
By a straightforward calculation and three integrations by parts, we get
where \(C_9>0\) as all subsequently appearing constants \(C_{10},C_{11},\dots \) is independent of \(p \ge 4\).
It is observed that by the Gagliardo–Nirenberg inequality, due to \(2 \le \frac{2(p+1)}{p}\le 2.5\) for \(p \ge 4\), one can pick \(C_{10}>1\) such that for all \(p\ge 4\),
Applying this together with the Young inequality, we obtain that for some \(C_{11}>0\),
where the fact that \(\frac{2(p+1)}{p}\le \frac{2p}{p-2}\le 4\) for any \(p\ge 4\) is used.
Similarly, we have
as well as
Consequently, inserting the above inequalities into (6.4.43) yields the existence of \(C_{12}>0\) such that
Now let \(p_k=4\cdot 2^k \) and \(M_k=\max \{1,\displaystyle \sup _{t\in (0,T_{max})}\int _\varOmega a^{p_k}(\cdot ,t)+b^{p_k}(\cdot ,t)+c^{p_k}(\cdot ,t)\}\) for \(k=0,1,2,\ldots \). Then (6.4.44) implies that for \(k=1,2,\ldots \)
which entails the existence of \(L>1\) independent of k such that
Therefore, by means of a standard recursive argument (see Pang and Wang 2018; Tao and Winkler 2014b for example), both when \(L^k M^4_{k-1}\le |\varOmega |(\Vert u_0\Vert _{L^\infty (\varOmega )}^{p_k}+\Vert w_0\Vert ^{p_k}_{L^\infty (\varOmega )}+\Vert z_0\Vert ^{p_k}_{L^\infty (\varOmega )}) \) for infinitely many \(k \ge 1\), and as well in the opposite case, we can obtain some \(C_{13}>0\) such that for all \(k\ge 1\)
from which, after taking \(k\rightarrow \infty \), the claims (6.4.34)–(6.4.36) readily follow.
According to Lemma 6.14, it remains for us to establish a priori estimates for \(\Vert \nabla v(\cdot ,t)\Vert _{L^4(\varOmega )}\).
Lemma 6.21
Let \(T>0\). Then there exists \(C(\hat{T})>0\) such that \(\Vert \nabla v(\cdot ,t)\Vert _{L^4(\varOmega )} \le C(\hat{T})\) for all \(t<\hat{T}\), where \(\hat{T}:=\min \{T, T_{max}\}\).
Proof
This can be achieved through an appropriate combination of three further testing processes, essentially relying on the \(L^\infty \)-estimates for a, b and c just asserted. We refrain from giving the proof and refer to Tao and Winkler (2020b) or Tao and Winkler (2014b) for details in a closely related setting.
We are now in the position to prove Theorem 6.2.
Proof of Theorem 6.2.  Thanks to the equivalence of (6.1.10) and (6.4.1) in the considered framework of classical solutions and in particular the extensibility criterion provided by Lemma 6.14, the proof is an evident consequence of Lemmas 6.20 and 6.21.
6.5 Asymptotic Behavior of Solutions to an Oncolytic Virotherapy Model
At the beginning of this subsection, in light of the method used in Horstmann and Winkler (2005) and Pang and Wang (2017), we provide the following statement on the local existence and extensibility of solutions to (6.1.12) as below.
Lemma 6.22
Suppose that \(\varOmega \subset \mathbb {R}^{N}\left( N\ge 1 \right) \) be a bounded domain with smooth boundary. Then one can find \(T_{max}\in (0,\infty ] \) and a unique quadruple of nonnegative functions \(\left( u,v,w,z \right) \in \left( C\left( \bar{\varOmega }\times [0,T_{max})\right) \bigcap C^{2,1}\left( \bar{\varOmega }\times (0,T_{max}) \right) \right) ^4\) which solves (6.1.12) classically in \(\varOmega \times (0,T_{max})\). Moreover, if \(T_{max}< +\infty \), then
for all \(q> \frac{N}{2}\).
To make the system mass-conserved, we introduce a nonnegative variable Q satisfying
where \(\frac{\rho _{u}+\rho _{z}}{\rho _{w}}\delta _{w}-\beta \ge 0\). Then it is easy to see that for all \(t\in (0,T)\),
which means
We first collect some easily verifiable observations in the following lemma. The constants \(c_{i},(i=1,2,3,4)\), \(c_{10}\) and \(C_{p_0}\) refer to Lemmas 1.1, 4.3 and 3.2 respectively.
Lemma 6.23
Under the assumptions of Theorem 6.3, there exist \(M_{i}> 1\) \(\left( i=0,1,\right. \left. \cdot \cdot \cdot ,6 \right) \) and \(\varepsilon (\xi _u,\xi _w,\xi _z,\rho _u,\rho _w,\rho _z,\alpha _y,\alpha _w) > 0\) such that
where
with \(\alpha \in (0,\min \left\{ \lambda _{1}, \delta _v\right\} )\) and \(\lambda _{1}> 0\) the first nonzero eigenvalue of \(-\varDelta \) in \(\varOmega \) under the Neumann condition.
For constants \(\alpha \in (0,\min \left\{ \lambda _{1}, \delta _v\right\} )\) and \(M_i>1( i=0,1,\ldots ,6 )\) referring to Lemma 6.23, let
By Lemma 6.22 and the smallness condition on the initial data in Theorem 6.3, \(T>0\) is well-defined. We first show \(T=T_{max}\). To this end, we will show that all of the estimates mentioned in (6.5.10) are valid with even smaller coefficients on the right-hand side. The derivation of these estimates will mainly rely on \(L^p-L^q\) estimates for the Neumann heat semigroup and the fact that the classical solutions on \((0,T_{max})\) can be represented as
for all \(t\in \left( 0,T_{max} \right) \) as per the variation-of-constants formula.
The global boundedness for solutions of (6.1.12) can be obtained directly from the following lemmas.
Lemma 6.24
Under the assumptions of Theorem 6.3, for all \(t\in \left( 0,T\right) \), we have
where \(M_0=1+M_6+2c_1\) with \(c_1\) defined in Lemma 1.1.
Proof
Set \(m_{\infty }=\frac{1}{\left| \varOmega \right| }\int _{\varOmega }\left( u_{0}+\frac{\rho _{u}+\rho _{z}}{\rho _{w}}w_{0}+z_{0} \right) \), then \(m_\infty \le \varepsilon \). It is obvious that
According to the Lemma 1.1(i), we know for all \(t\in \left( 0,T \right) \),
Then due to the definition of T and Lemma 1.1(i), we have
and hence end the proof.
Lemma 6.25
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\), we have
Proof
Applying (6.5.12), Lemma 1.1 (iii), we have
From Lemmas 1.1(ii) and 4.3, we obtain
From Hölder’s inequality, using Lemmas 1.1(iii) and 4.3, we get
Therefore, inserting the above two results into (6.5.16), we arrive at
According to (6.5.4), we thereby complete the proof.
Lemma 6.26
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\), we have
Proof
From (6.5.13), using Lemmas 1.1(iii) and 4.3, we have
which along with (6.5.5) implies that (6.5.17) is valid.
Similar as done in Lemma 6.26, we also have the following.
Lemma 6.27
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\), we have
Proof
From (6.5.14), using Lemmas 1.1(iii) and 4.3, we have
which together with (6.5.6) already implies that (6.5.18) holds.
Lemma 6.28
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\), we have
Proof
We know that
from which we obtain
It follows that
Noticing that
then
Similarly, we have
From Lemma 6.3, noticing \(p_0>1\), \(\alpha -\delta _v<0\), for all \(t\ge 0\), we obtain
Inserting (6.5.22) and (6.5.23) into (6.5.21) and using (6.5.24), we obtain
Therefore, (6.5.19) results from (6.5.7).
To obtain the estimate of \(\Vert \varDelta v(\cdot ,t)\Vert _{L^{p_0}(\varOmega )}\) for \(t\ge 0\), we need the following lemma.
Lemma 6.29
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\), we have
and
where
and
with \(k=\min \left\{ \frac{1}{2}(\delta _v-\alpha ),\delta _w\right\} \in (0,\alpha )\).
Proof
Denote \(G\left( x,t \right) =-\xi _{u}\nabla \cdot \left( u\nabla v \right) -\rho _{u}uz+\frac{1}{2}(\delta _v-\alpha ) u\), then \(u_t=\varDelta u-\frac{1}{2}(\delta _v-\alpha ) u+G(x,t)\) and
According to Lemma 3.2, we obtain
Similarly, denote \(F\left( x,t \right) =-\xi _{w}\nabla \cdot \left( w\nabla v \right) +\rho _{w}uz,\) then \(w_t=\varDelta w-\delta _ww+F(x,t)\) and
thus from Lemma 3.2, we obtain
and thereby complete the proof.
Lemma 6.30
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\),
Proof
By \(v\left( \cdot , t \right) =v_{0}\left( \cdot \right) e^{-\int _{0}^{t}\left( \alpha _{u}u+\alpha _{w}w+\delta _v \right) ds }\), we get
From (6.5.22) and (6.5.23), we obtain
and
From Lemma 6.3, noticing \(0<k<\delta _v-\alpha \), \(p_0>1\), we obtain
and
for all \(t\ge 0\), which together with Lemma 6.28 implies
Inserting (6.5.30), (6.5.31) and (6.5.32) into (6.5.29), we get
Therefore, (6.5.28) follows from (6.5.8).
Lemma 6.31
Under the assumptions of Theorem 6.3, for all \(t\in (0,T)\),
Proof
From Lemma 1.1(iv) and (6.5.11), using Lemma 4.3, it follows that
and in view of (6.5.9), we already arrive at (6.5.34) and complete the proof.
Proof of Theorem 6.3. First let us verify \(T=T_{max}\) by contraction. In fact, suppose that \(T< T_{max}\), then from Lemmas 6.25–6.31, it follows
for all \(t\in \left( 0,T \right) \), which contradicts the definition of T.
Next, we show that \(T_{max}=\infty \). In fact, if \(T_{max}<\infty \), then in view of the definition of T, we obtain
which contradicts with (6.5.1) in Lemma 6.22. Therefore, we have \(T_{max}=\infty \).
Integrating the equation of u in (6.1.12) over \(\varOmega \), we have
which along with the nonnegative property of u, z and the fact that \(\Vert u(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le M_0\varepsilon \) warrants that \(\bar{u}\left( t \right) :=\frac{1}{|\varOmega |}\int _{\varOmega }u\left( x,t \right) dx\) is noncreasing with respect to time t and and its limit \(t\rightarrow \infty \) exists, that is,
as well as
which implies
as \(t\rightarrow \infty \). On the other hand, by Poincare’s inequality, there exists \(k_1>0\) such that
By Embedding theorem, we know \(W^{1,2p_{0}}(\varOmega ) \hookrightarrow C^{1-\frac{N}{p_0}}(\overline{\varOmega })\), for \(p_{0}>\max \{1,\frac{N}{2}\}\). There exists \(k_{2}> 0\), such that
as \(t\rightarrow \infty \). Thus,
Now, we consider a linear combination of u and w
then
Accordingly,
Similarly, we obtain
and
Then
as \(t\rightarrow \infty \). Then denote \(w^*:=\frac{1}{\rho _u}(H^*-\rho _w u^*)\), as \(t\rightarrow \infty \), we obtain
Next, we consider a linear combination of u, w and z. Let
Then
Accordingly,
where \(\rho _{z}\delta _{w}-(\rho _u+\rho _w)\beta >0\).
Similarly, we obtain
and
Then
as \(t\rightarrow \infty \). Then denote \(z^*:=\frac{1}{\rho _u+\rho _w}(I^*-\rho _z(u^*+w^*))\), as \(t\rightarrow \infty \), we obtain
By contradiction, if \(w^*>0\) or \(z^*>0\), then there exists \(t^*>0\) such that for all \(t>t^*\),
which implies that \(\int _{\varOmega }Q(x,t)dx\rightarrow \infty \) as \(t\rightarrow \infty \) and thus contradicts with that \(\Vert Q(\cdot ,t)\Vert _{L^\infty (\varOmega )}\le M_0\varepsilon \). Hence, we have \(w^*=z^*=0\). On the other hand, by (6.5.20), it is easy to see that
So the proof of Theorem 6.3 is complete.
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Ke, Y., Li, J., Wang, Y. (2022). Multi-taxis Cross-Diffusion System. In: Analysis of Reaction-Diffusion Models with the Taxis Mechanism. Financial Mathematics and Fintech. Springer, Singapore. https://doi.org/10.1007/978-981-19-3763-7_6
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