Boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source

Abstract

In this paper, we deal with the following quasilinear attraction–repulsion model:

$$ \textstyle\begin{cases} u_{t}=\nabla \cdot (D(u)\nabla u)-\nabla \cdot ( S(u)\chi (v)\nabla v)+ \nabla \cdot ( F(u)\xi (w)\nabla w)+f(u), &x\in \varOmega , t>0, \\ v_{t}=\Delta v+\beta u-\alpha v, &x\in \varOmega ,t>0, \\ 0=\Delta w+\gamma u-\delta w, &x\in \varOmega , t>0, \\ u(x,0)=u_{0}(x), \quad\quad v(x,0)= v_{0}(x), &x\in \varOmega \end{cases} $$

with homogeneous Neumann boundary conditions in a smooth bounded domain $\varOmega \subset R^{n}$ ($n\geq 2$). Let the chemotactic sensitivity $\chi (v)$ be a positive constant, and let the chemotactic sensitivity $\xi (w)$ be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.

1 Introduction

In this paper, we consider a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source

$$ \textstyle\begin{cases} u_{t}=\nabla \cdot (D(u)\nabla u)-\nabla \cdot ( S(u)\chi (v)\nabla v)+ \nabla \cdot ( F(u)\xi (w)\nabla w)+f(u), &x\in \varOmega ,t>0, \\ v_{t}=\Delta v+\beta u-\alpha v, &x\in \varOmega ,t>0, \\ 0=\Delta w+\gamma u-\delta w, &x\in \varOmega ,t>0, \\ \frac{\partial u(x,t)}{\partial \nu }=\frac{\partial v(x,t)}{\partial \nu }=\frac{\partial w(x,t)}{\partial \nu }=0, &x\in \partial \varOmega ,t>0, \\ u(x,0)=u_{0}(x), \quad\quad v(x,0)= v_{0}(x), &x\in \varOmega , \end{cases} $$

(1.1)

where $\varOmega \subset R^{n}$ ($n\geq 2$) is a bounded domain with smooth boundary, and $\frac{\partial }{\partial \nu }$ denotes the derivative with respect to the outer normal of ∂Ω, α, β, γ, and δ are positive parameters, and $\chi (v)$ and $\xi (w)$ represent chemosensitivity. We assume that the functions $\chi (v)$ and $\xi (w)$ satisfy the following hypotheses:

$(H_{1})$ :

the function $\chi (v)=\chi _{0}$, which is a positive constant;

$(H_{2})$ :

the function $\xi (w)=\frac{\xi _{0}}{w}$ for all $w>0$, where $\xi _{0}$ is a positive constant.

Here $\chi _{0}$ is the strength of the attraction, and $\xi _{0}$ is the strength of the repulsion, $u(x,t)$, $v(x,t)$, and $w(x,t)$ denote the cell density, the concentration of the chemoattractant, and the concentration of the chemorepellent. We assume that

$$\begin{aligned} D(u),S(u),F(u)\in C^{2} \bigl([0,\infty )\bigr) \end{aligned}$$

(1.2)

and there exist constants $C_{D}>0$ and $m\geq 1$ such that

$$\begin{aligned} D(u)\geq C_{D}(u+1)^{m-1}. \end{aligned}$$

(1.3)

The function $f:[0,\infty )\rightarrow R$ is smooth and satisfies $f(0)\geq 0$ and

$$\begin{aligned} f(u)\leq a-bu^{\eta } \end{aligned}$$

(1.4)

with $a\geq 0$, $b>0$, and $\eta >1$. The initial data comply with

$$ \textstyle\begin{cases} u_{0}\in W^{1,\infty }(\varOmega ) \quad \text{with } u_{0}\geq 0 \text{ in } \varOmega \text{ and } u_{0}\not \equiv 0, \\ v_{0}\in W^{1,\infty }(\varOmega ) \quad \text{with }v_{0}\geq 0 \text{ in } \varOmega . \end{cases} $$

(1.5)

Chemotaxis describes the oriented movement of cells along the concentration gradient of a chemical signal produced by cells. The prototype of the chemotaxis model, known as the Keller–Segel model, was first proposed by Keller and Segel [3] in 1970:

$$ \textstyle\begin{cases} u_{t}=\Delta u-\nabla \cdot (u\chi (v)\nabla v), &x\in \varOmega , t>0, \\ v_{t}=\Delta v+u-v, &x\in \varOmega , t>0. \end{cases} $$

(1.6)

When $\chi (v)$ is a positive constant, a global solution is studied by Osaki and Yagi [8] for $n=1$; a global solution is investigated by Nagai et al. [7, 16] for $n\geq 2$; the blowup solutions are proved by Herrero ea al. [2, 12]. For the case where $\chi (v)\leq \frac{\chi _{0}}{(1+\alpha v)^{k}}$, $\alpha >0$, and $k>1$, the global classical solution is asserted by Winkler [17]. For the case $\chi (v)=\frac{\chi }{v}$ with a positive constant $\chi <\sqrt{\frac{2}{n}}$, a global classical solution is explored by Winkler [18].

Moreover, when $D(u)=1$ and $f(u)=0$, Tao and Wang [11] studied the following chemotaxis model:

$$ \textstyle\begin{cases} u_{t}=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (u\chi (v)\nabla v)+ \nabla \cdot (u\xi (w)\nabla w)+f(u), &x\in \varOmega , t>0, \\ 0=\Delta v+u-v, &x\in \varOmega , t>0, \\ 0=\Delta w+u-w, &x\in \varOmega , t>0. \end{cases} $$

(1.7)

The global boundedness of the solutions was obtained in high dimensions, and blowup solutions were identified in $R^{2}$.

In the case where $\chi (v)$ and $\xi (w)$ are positive parameters in (1.7), $D(u)$ satisfies (1.3), and $f(u)$ satisfies (1.4), a unique global bounded classical solution was deduced by Wang [15]. When $f(u)=0$ in (1.7), $\chi (v)$ and $\xi (w)$ are positive functions, $D(u)$ satisfies (1.3), and $f(u)$ satisfies (1.4), the global classical solutions are asserted by Wu and Wu [19], who obtained an important estimate of $\int _{\varOmega } \vert \nabla v \vert ^{2}\,dx$. Note that this method is not applicable for the general $f(u)$ in our paper. For more details about chemotaxis system, we refer the interested readers to [1, 5, 6, 9, 13, 14].

Motivated by [11, 15, 17,18,19], we consider a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source. Our main results are given as follows.

Theorem 1.1

Assume that (1.2)–(1.5), $(H_{1})$, and $(H_{2})$ are valid. Moreover, suppose that

$$\begin{aligned} 0\leq S(u)\leq C_{S}(u+1)^{s}, \quad\quad 0 \leq F(u)= C_{F}(u+1)^{\sigma }, \end{aligned}$$

(1.8)

and

$$ 0\leq s< \textstyle\begin{cases} \frac{m+\eta }{2}-\frac{n-1}{n}, & \eta \in (1,\frac{n+2}{n}], \\ \frac{m}{2}+\frac{\eta (n+4)}{2(n+2)}-1, & \eta \in (\frac{n+2}{n},n+2), \\ \frac{m+\eta }{2}, & \eta \in [n+2,\infty ). \end{cases} $$

• (i) If $\sigma \in (1,\eta )$, then (1.1) admits a bounded global classical solution.

• (ii) If $\sigma \in (\eta ,m)$, then (1.1) admits a bounded classical solution.

• (iii) If $m>\max \{1,\frac{n\sigma +2-2\sigma }{n+2}, \frac{n\sigma -2}{n}\}$, then (1.1) admits a bounded global classical solution.

The local existence and uniqueness of system (1.1) can be derived from Lemma 2.1 in [4], and hence we only state the result and omit its proof.

Lemma 1.1

([4])

Suppose that (1.2)–(1.5) are valid. Then there exist a maximal existence time $T_{\max }\in (0,+\infty )$ and a unique triplet (u,v,w) of functions that satisfy

$$ \textstyle\begin{cases} u\in C^{0}(\overline{\varOmega }\times [0,T_{\max }))\cap C^{2,1}(\overline{ \varOmega }\times [0,T_{\max })), \\ v\in C^{0}(\overline{\varOmega }\times [0,T_{\max }))\cap C^{2,1}(\overline{ \varOmega }\times [0,T_{\max }))\cap L^{\infty }((0,T_{\max });W^{1,l}( \varOmega )), \\ w\in C^{0}(\overline{\varOmega }\times [0,T_{\max }))\cap C^{2,1}(\overline{ \varOmega }\times [0,T_{\max }))\cap L^{\infty }((0,T_{\max });W^{1,l}( \varOmega )) \end{cases} $$

(1.9)

with $l>n$ and

$$\begin{aligned} u\geq 0, \quad\quad v\geq 0, \quad\quad w\geq 0 \quad \textit{in } \varOmega \times (0,T_{\max }). \end{aligned}$$

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

$$\begin{aligned} \lim_{t\rightarrow T_{\max }}\sup \bigl( \bigl\Vert u(\cdot ,t) \bigr\Vert \bigr)_{L^{\infty }}( \varOmega )+ \bigl\Vert v(\cdot ,t) \bigr\Vert _{w^{1,\infty }(\varOmega )}+ \bigl\Vert w( \cdot ,t) \bigr\Vert _{w ^{1,\infty }(\varOmega )}= \infty . \end{aligned}$$

(1.10)

Lemma 1.2

Let $(u,v,w)$ be the solution of system (1.1). Then there exist a constant $m^{*}$ such that

$$\begin{aligned} \int _{\varOmega }u(x,t)\,dx\leq m^{*}:=\max \biggl\{ \int _{\varOmega }u_{0}, \frac{a+b}{b} \vert \varOmega \vert \biggr\} , \quad t\in (0,T_{\max }). \end{aligned}$$

(1.11)

Proof

Integrating the first equation of system (1.1) over Ω, we have

$$\begin{aligned} \frac{d}{dt} \int _{\varOmega }u\,dx=a \vert \varOmega \vert -b \int _{\varOmega }u^{\eta }\,dx. \end{aligned}$$

(1.12)

Due to $\eta >1$ and Young’s inequality, we derive

$$\begin{aligned} u\leq u^{\eta }+1. \end{aligned}$$

(1.13)

Combining with (1.12), we have

$$\begin{aligned} \frac{d}{dt} \int _{\varOmega }u\,dx\leq -b \int _{\varOmega }u\,dx+(a+b) \vert \varOmega \vert , \end{aligned}$$

(1.14)

which yields (1.11). □

Lemma 1.3

(Gagliardo–Nirenberg inequality)

Let $r\in (0,\alpha )$ and $\psi \in W^{1,2}(\varOmega )\cap L^{r}(\varOmega )$. Then there exists a constant $C_{\mathrm{GN}}>0$ such that

$$\begin{aligned} \Vert \psi \Vert _{L^{\alpha }(\varOmega )}\leq C_{\mathrm{GN}} \bigl( \Vert \nabla \psi \Vert _{L^{2}(\varOmega )}^{\lambda ^{*}} \Vert \psi \Vert _{L^{r}(\varOmega )}^{1-\lambda ^{*}}+ \Vert \psi \Vert _{L^{r}(\varOmega )} \bigr) \end{aligned}$$

(1.15)

with

$$\begin{aligned} \lambda ^{*}=\frac{\frac{n}{r}-\frac{n}{\alpha }}{1-\frac{n}{2}+ \frac{n}{r}}\in (0,1). \end{aligned}$$

Lemma 1.4

Let Ω be a bounded domain in $R^{n}$ with smooth boundary, and let $v_{0}\in W^{1,\infty }(\varOmega )$. Suppose that there exists a constant $C_{1}$ such that

$$\begin{aligned} \Vert u \Vert _{L^{k}(\varOmega )}\leq C_{1}, \quad t\in (0,T). \end{aligned}$$

For the problem

$$ \textstyle\begin{cases} v_{t}=\Delta v+\beta u-\alpha v, &x\in \varOmega , t>0, \\ \frac{\partial v(x,t)}{\partial \nu }=0, &x\in \partial \varOmega , t>0, \end{cases} $$

• (i) if $1\leq k< n$, then

  $$\begin{aligned} \bigl\Vert v(t) \bigr\Vert _{W^{1,j}(\varOmega )}\leq C \quad \textit{for all } j \in \biggl(0,\frac{nk}{n-k} \biggr); \end{aligned}$$

  (1.16)

• (ii) if $k=n$, then (1.16) holds for all $j\in (0,\infty )$;

• (iii) if $k>n$, then (1.16) holds for $j=\infty$.

Lemma 1.5

([20])

For any $h\in [1,\frac{n}{n-1})$, there exists a constant $C_{2}>0$ such that

$$\begin{aligned} \bigl\Vert \nabla v(\cdot ,t) \bigr\Vert _{L^{h}}\leq C_{2}, \quad t\in (0,T_{\max }). \end{aligned}$$

(1.17)

Lemma 1.6

([21])

For any $h\in [1,\frac{n\eta }{(n+2-\eta )^{+}})$, there exists a constant $C_{3}>0$ such that

$$\begin{aligned} \bigl\Vert \nabla v(\cdot ,t) \bigr\Vert _{L^{h}}\leq C_{3}, \quad t\in (0,T_{\max }). \end{aligned}$$

(1.18)

2 A priori estimates

Lemma 2.1

Suppose

$$\begin{aligned}& \frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+\frac{d}{dt} \int _{\varOmega } \vert \nabla v \vert ^{2\beta } \,dx+D_{1} \int _{\varOmega } \bigl\vert \nabla (u+1)^{ \frac{k+m-1}{2}} \bigr\vert ^{2} \,dx \\& \quad\quad{} +\frac{bk}{2^{\eta +1}} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+ \int _{ \varOmega } \vert \nabla v \vert ^{2\beta }\,dx \\& \quad \leq D_{2} \int _{\varOmega }(u+1)^{k+\sigma -1}\,dx +D_{3}, \end{aligned}$$

(2.1)

where

$$ D_{1}=\frac{2C_{D}k(k-1)}{(k+m-1)^{2}}, \quad\quad D_{2}= \frac{C_{F}\xi _{0}k(k-1)}{k+\sigma -1}, \quad\quad D_{3} \textit{ is a constant}. $$

If $\sigma \in (1,\eta )$, then there exist constants $E_{1}>0$ and $E_{2}>0$ such that

$$\begin{aligned} \frac{d}{dt} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2 \beta }\,dx \biggr) +E_{1} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2\beta }\,dx \biggr) \leq E_{2} \end{aligned}$$

(2.2)

for sufficiently large k.

Proof

Since $\sigma \in (1,\eta )$, by Young’s inequality we have

$$\begin{aligned} \int _{\varOmega }(u+1)^{k+\sigma -1}\,dx\leq C_{4} \int _{\varOmega }(u+1)^{k+ \eta -1}\,dx+C_{5} \end{aligned}$$

(2.3)

and

$$\begin{aligned} \int _{\varOmega }(u+1)^{k}\,dx\leq C_{6} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+C _{7}. \end{aligned}$$

(2.4)

Combining (2.1), (2.3), and (2.4), we get that there are positive constants $E_{1}$ and $E_{2}$ such that

$$\begin{aligned} \frac{d}{dt} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2 \beta }\,dx \biggr) +E_{1} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2\beta }\,dx \biggr) \leq E_{2}. \end{aligned}$$

 □

Lemma 2.2

Suppose

$$\begin{aligned}& \frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+\frac{d}{dt} \int _{\varOmega } \vert \nabla v \vert ^{2\beta } \,dx+D_{1} \int _{\varOmega } \bigl\vert \nabla (u+1)^{ \frac{k+m-1}{2}} \bigr\vert ^{2} \,dx \\& \quad\quad{} +\frac{bk}{2^{\eta +1}} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+ \int _{ \varOmega } \vert \nabla v \vert ^{2\beta }\,dx \\& \quad \leq D_{2} \int _{\varOmega }(u+1)^{k+\sigma -1}\,dx +D_{3}, \end{aligned}$$

(2.5)

where

$$ D_{1}=\frac{2C_{D}k(k-1)}{(k+m-1)^{2}}, \quad\quad D_{2}= \frac{C_{F}\xi _{0}k(k-1)}{k+\sigma -1}, \quad\quad D_{3} \textit{ is a constant}. $$

If $\sigma \in (\eta ,m)$, then there exist constants $E_{3}>0$ and $E_{4}>0$ such that

$$\begin{aligned} \frac{d}{dt} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2 \beta }\,dx \biggr) +E_{3} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2\beta }\,dx \biggr) \leq E_{4}. \end{aligned}$$

(2.6)

Proof

By Lemma 1.2 and the Gagliardo–Nirenberg inequality there exists a constant $C_{8}>0$ such that

$$\begin{aligned}& \int _{\varOmega }(u+1)^{k+m-1}\,dx \\& \quad = \bigl\Vert (u+1)^{\frac{k+m-1}{2}} \bigr\Vert ^{2}_{L^{2}} \\& \quad \leq C_{\mathrm{GN}} \bigl( \bigl\Vert \nabla (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{2}}^{2\lambda ^{*}} \bigl\Vert (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{\frac{2}{k+m-1}}}^{2(1-\lambda ^{*})} + \bigl\Vert (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{\frac{2}{k+m-1}}}^{2} \bigr) \\& \quad \leq C_{8} \bigl( \bigl\Vert \nabla (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{2}}^{2\lambda ^{*}}+1 \bigr), \end{aligned}$$

(2.7)

where

$$ \lambda ^{*}=\frac{\frac{k+m-1}{2}-\frac{1}{2}}{\frac{k+m-1}{2}+ \frac{1}{n}-\frac{1}{2}}\in (0,1). $$

By Young’s inequality we obtain

$$\begin{aligned} \int _{\varOmega }(u+1)^{k+m-1}\,dx\leq C_{9} \int _{\varOmega } \bigl\vert \nabla (u+1)^{ \frac{k+m-1}{2}} \bigr\vert ^{2} \,dx+ C_{10}. \end{aligned}$$

(2.8)

Since $\sigma \in (\eta ,m)$, by Young’s inequality there exist $C_{11}>0$ and $C_{12}>0$ such that

$$\begin{aligned} \int _{\varOmega }(u+1)^{k+\sigma -1}\,dx\leq C_{11} \int _{\varOmega }(u+1)^{k+m-1}\,dx+C _{12}. \end{aligned}$$

(2.9)

Hence, combining (2.5), (2.8), and (2.9), we obtain (2.6). □

Lemma 2.3

Suppose

$$\begin{aligned}& \frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+\frac{d}{dt} \int _{\varOmega } \vert \nabla v \vert ^{2\beta } \,dx+D_{1} \int _{\varOmega } \bigl\vert \nabla (u+1)^{ \frac{k+m-1}{2}} \bigr\vert ^{2} \,dx \\& \quad\quad{} +\frac{bk}{2^{\eta +1}} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+ \int _{ \varOmega } \vert \nabla v \vert ^{2\beta }\,dx \\& \quad \leq D_{2} \int _{\varOmega }(u+1)^{k+\sigma -1}\,dx +D_{3}, \end{aligned}$$

(2.10)

where

$$ D_{1}=\frac{2C_{D}k(k-1)}{(k+m-1)^{2}}, \quad\quad D_{2}= \frac{C_{F}\xi _{0}k(k-1)}{k+ \sigma -1}, \quad\quad D_{3} \textit{ is a constant}. $$

If $m> \max \{1,\frac{n\sigma +2-2\sigma }{n+2},\frac{n\sigma -2}{n} \}$, then there exist constants $E_{5}>0$ and $E_{6}>0$ such that

$$\begin{aligned} \frac{d}{dt} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2 \beta }\,dx \biggr) +E_{5} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2\beta }\,dx \biggr) \leq E_{6}. \end{aligned}$$

(2.11)

Proof

By the Gagliardo–Nirenberg inequality there exists $C_{13}>0$ such that

$$\begin{aligned} \int (u+1)^{k+\sigma -1}\,dx ={} & \bigl\Vert (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{\frac{2(k+ \sigma -1)}{k+m-1}}}^{\frac{2(k+\sigma -1)}{k+m-1}} \\ \leq{} &C_{\mathrm{GN}} \bigl( \bigl\Vert \nabla (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{2}}^{\lambda _{1}} \bigl\Vert (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{\frac{2}{k+m-1}}}^{(1-\lambda _{1})} \\ &{}+ \bigl\Vert (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{\frac{2}{k+m-1}}}^{2} \bigr)^{\frac{2(k+ \sigma -1)}{k+m-1}} \\ \leq{} & C_{13} \bigl( \bigl\Vert \nabla (u+1)^{\frac{k+m-1}{2}} \bigr\Vert _{L^{2}}^{\lambda _{1}\cdot \frac{2(k+\sigma -1)}{k+m-1}}+1 \bigr), \end{aligned}$$

(2.12)

where

$$ \lambda _{1}=\frac{\frac{n(k+m-1)}{2}-\frac{n(k+m-1)}{2(k+\sigma -1)}}{1- \frac{n}{2}+\frac{n(k+m-1)}{2}}= \frac{n(k+m-1)(k+\sigma -1)-n(k+m-1)}{(k+ \sigma -1)[2-n+n(k+m-1)]}. $$

The condition $m>\max \{1,\frac{n\sigma +2-2\sigma }{n+2}\}$ and sufficiently large k guarantee that

$$\begin{aligned}& (k+f-1) \bigl[2-n+n(k+m-1) \bigr] \\& \quad =n(k+\sigma -1) (k+m-1)+(k+\sigma -1) (2-n) \\& \quad =n(k+\sigma -1) (k+m-1)+(k+m-1+\sigma -m) (2-n) \\& \quad \geq n(k+\sigma -1) (k+m-1)-n(k+m-1)+(2-n) (\sigma -m)+2(2m-1) \\& \quad \geq n(k+\sigma -1) (k+m-1)-n(k+m-1). \end{aligned}$$

Hence $\lambda _{1}\in (0,1)$.

Since $m>\max \{1,\frac{n\sigma -2}{n}\}$, we obtain

$$ \frac{k+\sigma -1 }{k+m-1}\cdot \lambda _{1}\in (0,1). $$

By Young’s inequality we derive

$$\begin{aligned} \int _{\varOmega }(u+1)^{k+m-1}\,dx\leq C_{14} \int _{\varOmega } \bigl\vert \nabla (u+1)^{ \frac{k+m-1}{2}} \bigr\vert ^{2} \,dx+ C_{15}. \end{aligned}$$

(2.13)

Therefore (2.10) and (2.13) yield (2.11). □

Lemma 2.4

Let $n\geq 2$. Defining

$$\begin{aligned} \lambda _{2}=\frac{2(k+\eta -1)}{\eta +m-2s}, \quad\quad \lambda _{3}= \frac{2(\beta -1)(k+\eta -1)}{k+\eta -3}, \end{aligned}$$

(2.14)

and

$$\begin{aligned} \delta _{i}(k,\beta ;h)=\frac{\frac{\beta }{h}-\frac{\beta }{\lambda _{i}}}{\frac{1}{n}-\frac{1}{2}+\frac{\beta }{h}}, \quad\quad w_{i}(k,\beta ;h)=\frac{\delta _{i}\lambda _{i}}{\beta }=\frac{\frac{ \lambda _{i}}{h}-1}{\frac{1}{n}-\frac{1}{2}+\frac{\beta }{h}}, \quad i=2,3, \end{aligned}$$

(2.15)

we have

• (a) if $\eta \in (1,\frac{n+2}{n}]$, $s<\frac{m+\eta }{2}- \frac{n-1}{n}$, then for sufficiently large k, there exist $\beta >2$ and $h\in [1,\frac{n}{n-1})$ such that

  $$\begin{aligned} \delta _{i}(k,\beta ;h)\in (0,1) \quad \textit{and} \quad w_{i}(k,\beta ;h)< 2, \quad i=2,3. \end{aligned}$$

  (2.16)

• (b) if $\eta \in (\frac{n+2}{n},n+2)$, $s<\frac{m}{2}+ \frac{\eta (n+4)}{2(n+2)}-1$, then for sufficiently large k, there exist $\beta >2$ and $h\in (\frac{n}{n-1},\frac{n\eta }{n+2-\eta })$ such that (2.16) holds.

Proof

By computation we verify that (2.16) is equivalent to

$$\begin{aligned} \lambda _{i}>h, \quad\quad \beta >\frac{\lambda _{i}}{2}- \frac{\lambda _{i}}{n}, \quad\quad \beta >\frac{ \lambda _{i}}{2}-\frac{h}{n}, \quad i=2,3. \end{aligned}$$

Thus it is sufficient to ensure that

$$\begin{aligned} \lambda _{i}>h, \quad\quad \beta >\frac{\lambda _{i}}{2}- \frac{h}{n}, \quad i=2,3. \end{aligned}$$

(2.17)

(a) For $h\in [1,\frac{n}{n-1}]$, by the continuity of h it suffices to prove the case $h=\frac{n}{n-1}$. To prove (2.17), we need to prove

$$\begin{aligned}& \frac{k+\eta -1}{\eta +m-2s}-\frac{1}{n-1}< \beta < \frac{k+\eta -1}{2}+ \frac{k+ \eta -3}{2(n-1)}, \end{aligned}$$

(2.18)

$$\begin{aligned}& k>\frac{n(m+n-2s)}{2(n-1)}+1-\eta , \quad\quad \beta > \frac{n(k+\eta -3)}{2(n-1)(k+\eta -1)}+1. \end{aligned}$$

(2.19)

Since $s<\frac{m+\eta }{2}-\frac{n-1}{n}$, there exists

$$ k>\max \biggl\{ 1, m+1-2s, 3-\eta , \frac{n(m+n-2s)}{2(n-1)}+1-\eta \biggr\} $$

such that

$$\begin{aligned} \frac{k+\eta -1}{\eta +m-2s}-\frac{1}{n-1}< \frac{k+\eta -1}{2}+\frac{k+ \eta -3}{2(n-1)}, \end{aligned}$$

so (2.18) and (2.19) are satisfied. Hence (2.17) holds.

(b) We note that $\eta \in (\frac{n+2}{n},n+2)$ ensures the interval $h\in (\frac{n}{n-1},\frac{n\eta }{n+2-\eta })$. By the continuity of h, let $h=\frac{n\eta }{n+2-\eta }$. To prove (2.17), we need to show that

$$\begin{aligned} \frac{k+\eta -1}{\eta +m-2s}-\frac{\eta }{n+2-\eta }< \beta < \frac{n+2}{2(n+2- \eta )} \cdot k-\frac{n+2}{2(n+2-\eta )}+\frac{n\eta }{2n+2-\eta } \end{aligned}$$

(2.20)

and

$$\begin{aligned} k>\frac{n\eta (\eta +m-2s)}{2(n+2-\eta )}-\eta +1, \quad\quad \beta > \frac{\eta (n-2)+2(n+2)}{2(n+2-\eta )}. \end{aligned}$$

(2.21)

Since $s<\frac{m}{2}+\frac{\eta (n+4)}{2(n+2)}-1$, there exists

$$ k>\max \biggl\{ 1, m+1-2s, 3-\eta , \frac{n\eta +2(n+2)}{2(n+2-\eta )} \cdot (\eta +m-2s)+1-\eta \biggr\} $$

such that

$$\begin{aligned} \frac{k+\eta -1}{\eta +m-2s}-\frac{\eta }{n+2-\eta }< \frac{n+2}{2(n+2- \eta )}\cdot k- \frac{n+2}{2(n+2-\eta )}+\frac{n\eta }{2n+2-\eta }. \end{aligned}$$

(2.22)

Then (2.20) and (2.21) are satisfied, and hence (2.17) holds. □

Lemma 2.5

For the second equation in (1.1), $E>0$, and $\beta >2$ we have

$$\begin{aligned} \begin{aligned}[b] & \frac{d}{dt} \int _{\varOmega } \vert \nabla v \vert ^{2\beta }\,dx + \frac{\beta -1}{ \beta } \int _{\varOmega } \bigl\vert \nabla \vert \nabla v \vert ^{\beta } \bigr\vert ^{2}\,dx\\&\quad \leq \bigl[4\beta (\beta -1)+ \beta n \bigr] \int _{\varOmega }u^{2} \vert \nabla v \vert ^{2\beta -2} \,dx+E\end{aligned} \end{aligned}$$

(2.23)

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

Proof

The proof can be found in [18]. □

Lemma 2.6

Under assumptions (1.2)–(1.5), $(H_{1})$, and $(H_{2})$, let $n\geq 2$ satisfy

$$ 0\leq s< \textstyle\begin{cases} \frac{m+\eta }{2}-\frac{n-1}{n} &\textit{for } \eta \in (1,\frac{n+2}{n}], \\ \frac{m}{2}+\frac{\eta (n+4)}{2(n+2)}-1 &\textit{for } \eta \in (\frac{n+2}{n},n+2), \\ \frac{m+\eta }{2} &\textit{for } \eta \in [n+2,\infty ), \end{cases} $$

and let $S(u)$ and $F(u)$ satisfy (1.8). If $\sigma \in (1, \eta )$, there exist sufficiently large k and $t\in [0,T_{\max })$ such that

$$\begin{aligned} \Vert u \Vert _{L^{k}(\varOmega )}\leq C. \end{aligned}$$

(2.24)

Proof

Multiplying by $(u+1)^{k-1}$ the both sides of the first equation in (1.1), we have

$$\begin{aligned} &\frac{1}{k}\frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+C_{D}(k-1) \int _{ \varOmega }(u+1)^{k+m-3} \vert \nabla u \vert ^{2}\,dx \\ &\quad \leq \chi _{0}(k-1) \int _{\varOmega }S(u) (u+1)^{k-2}\nabla u\cdot \nabla v \,dx \\ &\quad\quad{} -C_{F}(k-1) \int _{\varOmega }(u+1)^{f}\frac{\xi _{0}}{w}(u+1)^{k-2} \nabla u\cdot \nabla w\,dx \\ &\quad\quad{} +a \int _{\varOmega }(u+1)^{k-1}\,dx-b \int _{\varOmega }(u+1)^{k-1}u^{\eta }\,dx \end{aligned}$$

(2.25)

for all $t\in (0,T_{\max })$. Since $(u+1)^{\eta }\leq 2^{\eta -1}(u ^{\eta }+1)$ for $\eta >1$, this implies that

$$ u^{\eta }\geq \frac{1}{2^{\eta -1}}(u+1)^{\eta }-1. $$

Then (2.25) can be rewritten as

$$\begin{aligned} &\frac{1}{k}\frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+C_{D}(k-1) \int _{ \varOmega }(u+1)^{k+m-3} \vert \nabla u \vert ^{2}\,dx+\frac{b}{2^{\eta -1}} \int _{ \varOmega }(u+1)^{k+\eta -1}\,dx \\ &\quad \leq \chi _{0}(k-1) \int _{\varOmega }S(u) (u+1)^{k-2}\nabla u\cdot \nabla v\,dx \\ &\quad \quad{} -C_{F}(k-1) \int _{\varOmega }(u+1)^{f}\frac{\xi _{0}}{w}(u+1)^{k-2} \nabla u\cdot \nabla w\,dx +(a+b) \int _{\varOmega }(u+1)^{k-1}\,dx \\ &\quad = I_{1}+I_{2}+I_{3}, \end{aligned}$$

(2.26)

where

$$\begin{aligned} I_{1} &=\chi _{0}(k-1) \int _{\varOmega }S(u) (u+1)^{k-2}\nabla u\cdot \nabla v\,dx \\ &\leq \chi _{0}C_{S}(k-1) \int _{\varOmega }(u+1)^{k+s-2} \vert \nabla u \vert \vert \nabla v \vert \,dx \\ &\leq \frac{C_{D}(k-1)}{2} \int _{\varOmega }(u+1)^{k+m-3} \vert \nabla u \vert ^{2}\,dx \\ &\quad{} +\frac{ \chi _{0}^{2}C_{S}^{2}(k-1)}{2C_{D}} \int _{\varOmega }(u+1)^{k+2s-m-1} \vert \nabla v \vert ^{2}\,dx. \end{aligned}$$

(2.27)

Similarly, we have

$$ I_{2} =-C_{F}(k-1) \int _{\varOmega }(u+1)^{k+f-2}\frac{\xi _{0}}{w}\nabla u\nabla w\,dx =\frac{C_{F}(k-1)\xi _{0}}{k+f-1} \int _{\varOmega }(u+1)^{k+f-1} \nabla \cdot \biggl( \frac{1}{w}\nabla w\,dx \biggr), $$

and then

$$\begin{aligned} I_{2} &=\frac{C_{F}(k-1)\xi _{0}}{k+f-1} \int _{\varOmega }(u+1)^{k+f-1} \biggl(-\frac{1}{w ^{2}} \vert \nabla w \vert ^{2}+\frac{1}{w}\Delta w \biggr)\,dx \\ &\leq \frac{C_{F}(k-1)\xi _{0}}{k+f-1} \int _{\varOmega }(u+1)^{k+f-1} \frac{1}{w}(\delta w- \gamma u)\,dx \\ &\leq \frac{C_{F}(k-1)\xi _{0}}{k+f-1}\delta \int (u+1)^{k+f-1}\,dx. \end{aligned}$$

(2.28)

For all $t\in (0,T_{\max })$ with $C_{16}>0$, we obtain

$$\begin{aligned} I_{3}(a+b) &= \int _{\varOmega }(u+1)^{k-1}\,dx \\ &\leq \frac{b}{2^{\eta }} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+C_{16}. \end{aligned}$$

(2.29)

Combining (2.23), (2.27), (2.28), (2.29), and Young’s inequality, we deduce

$$\begin{aligned} &\frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+\frac{d}{dt} \int _{\varOmega } \vert \nabla v \vert ^{2\beta }\,dx+ \frac{2C_{D}k(k-1)}{(k+m-1)^{2}} \int _{\varOmega } \bigl\vert \nabla (u+1)^{\frac{k+m-1}{2}} \bigr\vert ^{2} \,dx \\ &\quad\quad {} +\frac{bk}{2^{\eta }} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx +\frac{\beta -1}{\beta } \int _{\varOmega } \bigl\vert \nabla \vert \nabla v \vert ^{\beta } \bigr\vert ^{2}\,dx \\ &\quad \leq \frac{\chi _{0}^{2}C_{s}^{2}k(k-1)}{2C_{D}} \int _{\varOmega }(u+1)^{k+2s-m-1} \vert \nabla v \vert ^{2}\,dx+\frac{C_{F}\xi _{0}k(k-1)}{k+f-1}\delta \int _{\varOmega }(u+1)^{k+f-1}\,dx+C _{16} \\ &\quad\quad{} + \bigl[4\beta (\beta -1)+\beta n \bigr] \int _{\varOmega }u^{2} \vert \nabla v \vert ^{2\beta -2} \,dx+E \\ &\quad \leq \frac{bk}{2^{\eta +1}} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+C_{17} \int _{\varOmega } \vert \nabla v \vert ^{\lambda _{2}} \,dx+C_{18} \int _{\varOmega } \vert \nabla v \vert ^{\lambda _{3}}\,dx \\ &\quad\quad{} +\frac{C_{F}\xi _{0}k(k-1)}{k+f-1}\delta \int _{\delta }(u+1)^{k+f-1}\,dx +C_{19} \end{aligned}$$

(2.30)

with $C_{17}, C_{18}, C_{19}>0$ and $\lambda _{2}$, $\lambda _{3}$ as shown in Lemma 2.4 for all $t\in (0,T_{\max })$. By Lemma 1.5, Lemma 1.6, and the Gagliardo–Nirenberg inequality we have

$$\begin{aligned} & \int _{\varOmega } \vert \nabla v \vert ^{\lambda _{i}}\,dx \\ &\quad = \bigl\Vert \vert \nabla v \vert ^{\beta } \bigr\Vert _{L^{\frac{\lambda _{i}}{\beta }}}^{\frac{ \lambda _{i}}{\beta }} \\ &\quad \leq C_{20} \bigl( \bigl\Vert \nabla \vert \nabla v \vert ^{\beta } \bigr\Vert _{L^{2}}^{\delta _{i}} \bigl\Vert \vert \nabla v \vert ^{\beta } \bigr\Vert _{L^{\frac{h}{\beta }}}^{1-\delta _{i}} + \bigl\Vert \vert \nabla v \vert ^{\beta } \bigr\Vert _{L^{\frac{h}{\beta }}} \bigr)^{\frac{\lambda _{i}}{\beta }} \\ &\quad \leq C_{21} \bigl\Vert \nabla \vert \nabla v \vert ^{\beta } \bigr\Vert _{L_{2}}^{\frac{\delta _{i}\lambda _{i}}{\beta }}+C_{22} \end{aligned}$$

with $\lambda _{i}$, $\delta _{i}$ as in Lemma 2.4, where $i=2,3$. Since $w_{i}=\frac{\delta _{i}\lambda _{i}}{\beta }<2$, by Young’s inequality we have

$$\begin{aligned} \int _{\varOmega } \vert \nabla v \vert ^{\lambda _{i}}\,dx\leq C_{23} \int _{\varOmega } \bigl\vert \nabla \vert \nabla v \vert ^{\beta } \bigr\vert ^{2}+C_{24}. \end{aligned}$$

(2.31)

From (2.30) and (2.31) we have that there exist constants $D_{1},D_{2},D_{3}>0$ such that

$$\begin{aligned} &\frac{d}{dt} \int _{\varOmega }(u+1)^{k}\,dx+\frac{d}{dt} \int _{\varOmega } \vert \nabla v \vert ^{2\beta } \,dx+D_{1} \int _{\varOmega } \bigl\vert \nabla (u+1)^{ \frac{k+m-1}{2}} \bigr\vert ^{2} \,dx \\ &\quad\quad{} +\frac{bk}{2^{\eta +1}} \int _{\varOmega }(u+1)^{k+\eta -1}\,dx+ \int _{ \varOmega } \vert \nabla v \vert ^{2\beta }\,dx \\ &\quad \leq D_{2} \int _{\varOmega }(u+1)^{k+\sigma -1}\,dx +D_{3}, \end{aligned}$$

(2.32)

where $D_{1}=\frac{2C_{D}k(k-1)}{(k+m-1)^{2}}$ and $D_{2}=\frac{C_{F} \xi _{0}k(k-1)}{k+\sigma -1}$. By Lemma 2.1 we have

$$\begin{aligned} \begin{aligned}[b] & \frac{d}{dt} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{\varOmega } \vert \nabla v \vert ^{2 \beta }\,dx \biggr) +C_{25} \biggl( \int _{\varOmega }(u+1)^{k}\,dx+ \int _{ \varOmega } \vert \nabla v \vert ^{2\beta }\,dx \biggr) \\&\quad \leq C_{26}\end{aligned} \end{aligned}$$

(2.33)

for all $t\in (0,T_{\max })$. By an ODE comparison argument we obtain (2.24).

For $\eta \in [n+2,\infty )$, from the Lemma 1.6 we have

$$\begin{aligned} \int _{\varOmega } \vert \nabla v \vert ^{\lambda _{i}}\,dx\leq C. \end{aligned}$$

(2.34)

In addition, $s<\frac{m+\eta }{2}$ is equivalent to $k+2s-m-1< k+ \eta -1$, so by (2.30), (2.34), and Lemma 2.1, using an ODE comparison argument, we derive (2.24). □

Remark 2.1

If $\sigma \in (\eta ,m)$ in Theorem 1.1, then by (2.32), Lemma 2.2, and Lemma 2.4 we obtain (2.24).

Remark 2.2

If $m>\max \{1,\frac{n\sigma +2-2\sigma }{n+2},\frac{n\sigma -2}{n}\}$ in Theorem 1.1, then by (2.32), Lemma 2.3, and Lemma 2.4 we obtain (2.24).

Proof of Theorem 1.1

For $k>\frac{n}{2}$, by Lemmas 1.4 and 2.6 there exists a positive constant $C_{27}$ such that

$$\begin{aligned} \bigl\Vert v(\cdot ,t) \bigr\Vert _{W^{1,\infty }(\varOmega )}\leq C_{27}. \end{aligned}$$

Using the elliptic regularity theory, we have

$$\begin{aligned} \bigl\Vert w(\cdot ,t) \bigr\Vert _{w^{2,k}(\varOmega )}\leq C_{28}. \end{aligned}$$

(2.35)

Then, for a sufficiently large k, by the Sobolev embedding theorem there exists a positive constant $C_{29}$ such that

$$\begin{aligned} \bigl\Vert \nabla w(\cdot ,t) \bigr\Vert _{L^{\infty }(\varOmega )}\leq C_{29}. \end{aligned}$$

(2.36)

By using Lemma A.1 in [10] we conclude that u is uniformly bounded in $\varOmega \times (0,T_{\max })$. Thus there exists a positive constant $C_{30}$ such that

$$\begin{aligned} \bigl\Vert u(\cdot ,t) \bigr\Vert _{L^{\infty }}\leq C_{30}, \quad t\in (0,T_{\max }), \end{aligned}$$

(2.37)

that is, $(u,v,w)$ is a global bounded classical solution to (1.1). □