Global Weak Solution in a p-Laplacian Attraction-Repulsion Chemotaxis System with Nonlinear Sensitivity and Signal Production

1. Introduction

In this paper, we study the following p-Laplacian chemotaxis system with nonlinear sensitivity and signal production:

$$\left\{\begin{array}{c}{u}_t=\nabla ·{(|\nabla u|}^{p-2}\nabla u)-\chi \nabla ·(u^{\alpha }\nabla v)+\xi \nabla ·(u^{\beta }\nabla w)+f(u),x\in \Omega ,t>0,\hfill \\ v_t=\Delta v-\sigma v+g_1(u),x\in \Omega ,t>0,\hfill \\ 0=\Delta w-\delta w+g_2(u),x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0,x\in \partial \Omega ,t>0\hfill \\ u(x,0)=u_0(x),v(x,0)=v_0(x),w(x,0)=w_0(x),x\in \Omega ,\hfill \end{array}\right.$$

(1.1)

where $\chi ,\xi ,\alpha ,\beta ,\eta ,\sigma ,\gamma ,\delta ,k_1,k_2>0$ and $p\ge 2$, $\Omega \subset {\mathbb{R}}^n(n\ge 2)$ is a smoothly bounded domain. u is the density of the cells, $v,w$ denote the concentration of the chemoattractant and chemorepellant. The nonlinear diffusion ${\Delta }_{p}u=▽·{(|\nabla u|}^{p-2}\nabla u)$ is called the slow and fast $p-$Laplacian diffusion if $p>2$ and $1<p<2$, respectively. The smooth logistic-type source $f:\mathbb{R}\to \mathbb{R}$ satisfies for all s>0,

$$\begin{array}{c}f(s)\le \kappa -\mu s^m\mathrm{and}f(0)\ge 0\hfill \end{array}$$

(1.2)

with $\kappa \in \mathbb{R},\mu >0$ and $m>1$. The signal production $g_i(i=1,2)$ satisfies for all s>0,

$$\begin{array}{c}{g}_1(s)=\eta s^{k_1},g_2(s)=\gamma s^{k_2},\hfill \end{array}$$

(1.3)

where $\eta ,\gamma ,k_1,k_2>0$.

Considering the effect of attraction without repulsion, problem (1.1) becomes the following Keller-Segel chemotaxis system

$$\left\{\begin{array}{c}{u}_t=▵u-\chi ·(u\nabla v),x\in \Omega ,t>0,\hfill \\ v_t=▵v-v+g(u),x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}=0,x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=u_0(x),v(x,0)=v_0(x),x\in \Omega .\hfill \end{array}\right.$$

(1.4)

When the signal production $g(u)$ is linear (i.e.$g(u)=u$), it was showed that the solutions of (1.4) are global bounded when $n=1$, or $n=2$ with ${\int }_{\Omega }{u}_0<{\displaystyle \frac{4\pi }{\chi }}$, or $n\ge 3$ with small initial conditions(see [1,19,23,35]), whereas when $n=2$ with ${\int }_{\Omega }{u}_0>{\displaystyle \frac{4\pi }{\chi }}$, or $n\ge 3$, the finite-time blow up may occur for a large class of initial date [2,34]. When the signal production $g(u)$ is nonlinear (i.e.$0<g(u)\le u^k$), the boundedness of (1.4) was studied in [17] if $k\in (0,{\displaystyle \frac{2}{n}})$ with $n\ge 2$. Considering the solution of (1.4) with nonlinear production $g(u)=u{(u+1)}^{k-1}$ under the condition $f(u)=u-\mu u^m$, it was proved in [20,21] that the weak solution of (1.4) is bounded globally if ${\displaystyle \frac{2(n+4)}{n+6}}<m\le 2$, $0<k<{\displaystyle \frac{(n+6)(m-1)}{2(n+2)}}$ and $n=2,3$. Recently, Zhuang et al. [39] got the globally bounded classical solution of (1.4) under $k<m-1$, or $k=m-1$ with $\mu >0$ sufficiently large.

Now, recall a more general attraction-repulsion chemotaxis system

$$\left\{\begin{array}{c}{u}_t=\nabla ·(D(u)\nabla u)-\chi \nabla ·(u\nabla v)+\xi \nabla ·(u\nabla w)+f(u),x\in \Omega ,t>0,\hfill \\ {\tau }_1{v}_t=▵v-\sigma v+g_1(u),x\in \Omega ,t>0,\hfill \\ {\tau }_2{w}_t=▵w-\delta w+g_2(u),x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}={\displaystyle \frac{\partial v}{\partial \nu }}={\displaystyle \frac{\partial w}{\partial \nu }}=0,x\in \partial \Omega ,t>0\hfill \\ u(x,0)=u_0(x),v(x,0)=v_0(x),w(x,0)=w_0(x),x\in \Omega ,\hfill \end{array}\right.$$

(1.5)

where ${\tau }_i=\{0,1\}(i=1,2)$. For the case $f(u)\equiv 0,g_1(u)=\eta u$, $g_2(u)=\gamma u$. If ${\tau }_1=1$, ${\tau }_2=0$ and $D(u)$ satisfying $D(u)=D_0{u}^{\theta }$ ,there exists a global weak solution under the case $\theta >1-{\displaystyle \frac{2}{n}}$[14]. Later, [8] optimized the condition of $\theta$ into $\theta >1-{\displaystyle \frac{4}{n+2}}$ but added $\xi \gamma -\chi \eta \ge 0$. If ${\tau }_1={\tau }_2=0$, and considering $p-$Laplacian diffusion (i.e. $D(u)={|\nabla u|}^{p-2}$), it was showed in [12] that the system (1.5) exists a global bounded weak solution under the case (i) $\xi \gamma -\chi \eta \le 0$, or case (ii) $\xi \gamma -\chi \eta >0$ with $p>{\displaystyle \frac{3n}{n+1}}$, or $1<p\le {\displaystyle \frac{3n}{n+1}}$ and $\parallel u_0{\parallel }_{L{\displaystyle \frac{(3-p)n}{p}}}$ is small. More results on attraction-repulsion chemotaxis system can be found in [1,6,7,9,10,11,24,26,29,31,36,38]. For the case $f(u)=\kappa -\mu u^m,g_1(u)=\eta u^{k_1},g_2(u)=\gamma u^{k_2}$. If ${\tau }_1=1$, ${\tau }_2=0$, it was shown that when (i) $m>max\{2k_1,{\displaystyle \frac{2k_{1}n}{2+n}}+{\displaystyle \frac{1}{p-1}}\}$, or case (ii) $k_2>max\{2k_1-1,{\displaystyle \frac{2k_{1}n}{2+n}}+{\displaystyle \frac{2-p}{p-1}}\}$ with $m>2$, the problem (1.5) possesses a global bounded weak solution [32]. If ${\tau }_1={\tau }_2=1$, Jia [3] proved that when $max\{k_1,k_2\}<m-1$ or $max\{k_1,k_2\}=m-1$ with large $\mu >0$, there exists a global bounded weak solution. More results on p-Laplacian chemotaxis models can be found in [4,12,15,16,18,27,28,37,40].

We now formulate the principal result .

Theorem 1.1.

Let $p\ge 2$ and $\Omega \subset {\mathbb{R}}^n(n\ge 2)$ be a bounded domain with smooth boundary. Assume that (1.2), (1.3) hold with $k_1>0$, $0<k_2\le 1$ and $m>1$. Then for any non-negative initial data $u_0,v_0,w_0\in W^{1,\infty }(\Omega )$, problem (1.1) has a globally bounded weak solution $(u,v,w)\in L^{\infty }(\Omega )\times W^{1,\infty }(\Omega )\times W^{1,\infty }(\Omega )$ under the assumption (i) $m>max\{2k_1,{\displaystyle \frac{p\alpha }{p-1}}-1\}$, or (ii) $k_2>max\{2k_1-\beta ,{\displaystyle \frac{p\alpha }{p-1}}-\beta -1\}$ with $m>max\{2,2\alpha ,2\beta \}$.

Remark 1.1.

When $\alpha =\beta =1$ for (1.1), we optimized the assumption (i) $m>max\{2k_1,{\displaystyle \frac{2k_{1}n}{2+n}}+{\displaystyle \frac{1}{p-1}}\}$, or (ii) $k_2>max\{2k_1-1,{\displaystyle \frac{2k_{1}n}{2+n}}+{\displaystyle \frac{2-p}{p-1}}\}$ with $m>2$ of reference [32] into (i) $m>max\{2k_1,{\displaystyle \frac{1}{p-1}}\}$ or (ii) $k_2>max\{2k_1-1,{\displaystyle \frac{2-p}{p-1}}\}$ with $m>2$.

This paper is organized as follows. In Sec. 2, we will introduce a regularized problem (2.1) and preliminary lemmas. In Sec. 3, we prove the boundedness of weak solutions of (2.1). Finally, we prove Theorem 1.1 by an approximation procedure in Sec. 4.

2. Preliminaries

Due to the p-Laplacian diffusion, we begin with the definition of weak solution to (1.1).

Definition 2.1.

Let

$$0\le u\in L_{loc}^2([0,T);L^2(\Omega )),$$

$$0\le v,w\in L_{loc}^2([0,T);W^{1,2}(\Omega ))$$

We call $(u,v,w)$ a weak solution of (1.1) if the following equalities hold:

$$\begin{array}{c}-{\int }_0^T{\int }_{\Omega }u{\phi }_t-{\int }_{\Omega }{u}_0(x)\phi (x,0)=-{\int }_0^T{\int }_{\Omega }{|\nabla u|}^{p-2}\nabla u·\nabla \phi +\chi {\int }_0^T{\int }_{\Omega }{u}^{\alpha }\nabla v·\nabla \phi \hfill \\ -\xi {\int }_0^T{\int }_{\Omega }{u}^{\beta }\nabla w·\nabla \phi +{\int }_0^T{\int }_{\Omega }f(u)\phi ,\hfill \end{array}$$

$$\begin{array}{c}-{\int }_0^T{\int }_{\Omega }v{\phi }_t-{\int }_{\Omega }{v}_0(x)\phi (x,0)=-{\int }_0^T{\int }_{\Omega }\nabla v·\nabla \phi -\sigma {\int }_0^T{\int }_{\Omega }v\phi +{\int }_0^T{\int }_{\Omega }{g}_1(u)\phi \hfill \end{array}$$

and

$$\begin{array}{c}0=-{\int }_0^T{\int }_{\Omega }\nabla w·\nabla \phi -\delta {\int }_0^T{\int }_{\Omega }w\phi +{\int }_0^T{\int }_{\Omega }{g}_2(u)\phi \hfill \end{array}$$

for all $\phi \in C_0^{\infty }(\overline{\Omega }\times [0,T))$.

In order to construct weak solutions of (1.1), we consider the regularized problem

$$\left\{\begin{array}{c}{u}_{\epsilon t}=\nabla ·((|\nabla u_{\epsilon }{|}^2+\epsilon {)}^{{\displaystyle \frac{p-2}{2}}}\nabla u_{\epsilon })-\chi \nabla ·(u_{\epsilon }^{\alpha }\nabla v_{\epsilon })+\xi \nabla ·(u_{\epsilon }^{\beta }\nabla w_{\epsilon })+f(u_{\epsilon }),\hfill \\ v_{\epsilon t}=▵v_{\epsilon }-\sigma v_{\epsilon }+g_1(u_{\epsilon }),x\in \Omega ,t>0,\hfill \\ 0=▵w_{\epsilon }-\delta w_{\epsilon }+g_2(u_{\epsilon }),x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial u_{\epsilon }}{\partial \nu }}={\displaystyle \frac{\partial v_{\epsilon }}{\partial \nu }}={\displaystyle \frac{\partial w_{\epsilon }}{\partial \nu }}=0,x\in \partial \Omega ,t>0\hfill \\ u_{\epsilon }(x,0)=u_0(x),v_{\epsilon }(x,0)=v_0(x),w_{\epsilon }(x,0)=w_0(x),x\in \Omega ,\hfill \end{array}\right.$$

(2.1)

for each $\epsilon \in (0,1)$. System (2.1) is locally solvable in the classical sense by using a fixed point theorem similar to [27,29].

Lemma 2.1.

Let $p\ge 2$ and $\Omega \subset {\mathbb{R}}^n(n\ge 2)$ be a smooth bounded domain, and that the initial data value $u_0,v_0,w_0\in W^{1,\infty }(\Omega )$. Then for any $\epsilon \in (0,1)$, there exists $T_{max,\epsilon }\in (0,\infty ]$ and a local-in-time classical solution $(u_{\epsilon },v_{\epsilon },w_{\epsilon })\in C(\overline{\Omega }\times [0,T_{max,\epsilon }))\cap C^{2,1}(\overline{\Omega }\times [0,T_{max,\epsilon }))$ that satisfies (2.1). Moreover, if $T_{max,\epsilon }<\infty$, then

$$\begin{array}{c}\underset{t\to T_{max,\epsilon }}{lim}{\parallel u_{\epsilon }(·,t)\parallel }_{L^{\infty }(\Omega )}=\infty .\hfill \end{array}$$

Lemma 2.2.

Assume (1.2) holds, then there exists a positive constant $M:=M(u_0,|\Omega |)$ such that

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon }(x,t)dx\le Mfort\in (0,T_{max,\epsilon }).\hfill \end{array}$$

(2.2)

Proof. 

Integrating the first functions of (2.1), we have by using (1.2) that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{u}_{\epsilon }dx\le \kappa |\Omega |-{\mu |\Omega |}^{1-m}{({\int }_{\Omega }{u}_{\epsilon }dx)}^m,t\in (0,T_{max,\epsilon })\hfill \end{array}$$

(2.3)

which implies (2.2).□

Lemma 2.3.

For any ${\epsilon }_1,{\epsilon }_2>0$ and $q>0$, there exists constants $C_1,C_2>0$ such that the solution z of the problem

$$\begin{array}{c}-▵z+\delta z=\gamma u^{k_2}in\Omega ,{\displaystyle \frac{\partial z}{\partial \nu }}=0on\partial \Omega ,\hfill \end{array}$$

(2.4)

satisfies

$$\begin{array}{c}{\int }_{\Omega }{z}^{q+1}\le {\epsilon }_1{\int }_{\Omega }{u}^{k_2(q+1)}+C_1\le {\epsilon }_2{\int }_{\Omega }{u}^{q+1}+C_2.\hfill \end{array}$$

(2.5)

for $u\in L^1(\Omega )$ and $0<k_2\le 1$.

Proof. 

The proof closely resembles that of Lemma 2.2 in [30]; therefore, we will omit it.□

Next, we present the Gagliardo-Nirenberg inequality, which will be frequently utilized in the subsequent analysis [13,22].

Lemma 2.4.

Assume that $1\le p,q\le \infty$ satisfying $(n-q)p\le nq$. Let $s>0$ and $0<r\le p\le \infty$. Then for any $\varphi \in W^{1,q}(\Omega )\cap L^r(\Omega )$, there exists a constant $C_{GN}>0$ such that

$$\begin{array}{c}{\parallel \varphi \parallel }_{L^p(\Omega )}\le C_{GN}{(\parallel \nabla \varphi \parallel }_{L^q(\Omega )}^{{\lambda }^{*}}{\parallel \varphi \parallel }_{L^r(\Omega )}^{1-{\lambda }^{*}}+{\parallel \varphi \parallel }_{L^s(\Omega )}),\hfill \end{array}$$

(2.6)

where

$$\begin{array}{c}{\lambda }^{*}={\displaystyle \frac{\frac{n}{r}-\frac{n}{p}}{1-\frac{n}{q}+\frac{n}{r}}}\in (0,1).\hfill \end{array}$$

(2.7)

3. Regularity Estimates to Regularized Problem

In the section, we study the global boundedness to the regularized problem (2.1), we pay attention to the following lemma.

Lemma 3.1.

Assume that (1.2), (1.3) hold with $k_1>0$, $0<k_2\le 1$ and $m>1$. If $m>2k_1$ or $k_2>2k_1-\beta$ with $m>max\{2,2\alpha ,2\beta \}$, there exists a constant $C_1>0$ such that

$$\begin{array}{c}{\int }_{\Omega }{|\nabla v_{\epsilon }|}^{2}dx\le C_{1}ont\in (0,T_{max}).\hfill \end{array}$$

(3.1)

for any $\epsilon \in (0,1)$.

Proof. 

Case (i): $m>2k_1$. Multiplying ${(2.1)}_2$ by $-2▵v_{\epsilon }$, integrating by parts, we obtain

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }|\nabla v_{\epsilon }{|}^2=-2{\int }_{\Omega }|▵v_{\epsilon }{|}^2-2\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2-2\eta {\int }_{\Omega }{u}_{\epsilon }^{k_1}▵v_{\epsilon }\hfill \\ \le -2{\int }_{\Omega }|▵v_{\epsilon }{|}^2-2\sigma {\int }_{\Omega }|\nabla v_{\epsilon }{|}^2+2{\int }_{\Omega }{|▵v_{\epsilon }|}^2+{\displaystyle \frac{{\eta }^2}{2}}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}\hfill \\ =-2\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2+{\displaystyle \frac{{\eta }^2}{2}}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}\hfill \end{array}$$

(3.2)

Together with (3.2) and the first equation of (2.1), there exists a constant $C_2>0$ such that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}({\int }_{\Omega }{u}_{\epsilon }+{\int }_{\Omega }|\nabla v_{\epsilon }{|}^2)\le \kappa |\Omega |-\mu {\int }_{\Omega }{u}_{\epsilon }^m-2\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2+{\displaystyle \frac{{\eta }^2}{2}}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}\hfill \\ \le -2\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2+C_2,\hfill \end{array}$$

(3.3)

which used $m>2k_1$ and Young’s inequality. Collecting Lemma 2.2 and (3.3), we have

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}({\int }_{\Omega }{u}_{\epsilon }+{\int }_{\Omega }|\nabla v_{\epsilon }{|}^2)+2\sigma ({\int }_{\Omega }{u}_{\epsilon }+{\int }_{\Omega }|\nabla v_{\epsilon }{|}^2)\le 2\sigma {\int }_{\Omega }{u}_{\epsilon }+C_2\le C_3,\hfill \end{array}$$

(3.4)

Case (ii): $k_2>2k_1-\beta$ with $m>max\{2,2\alpha ,2\beta \}$. Multiplying the first equation of (2.1) by $1+ln{u}_{\epsilon }$ and using Young’s inequality, we get

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{u}_{\epsilon }ln{u}_{\epsilon }\le -{({\displaystyle \frac{p}{p-1}})}^p{\int }_{\Omega }{|\nabla u_{\epsilon }^{{\displaystyle \frac{p-1}{p}}}|}^p+\chi {\int }_{\Omega }{u}_{\epsilon }^{\alpha -1}\nabla u_{\epsilon }·\nabla v_{\epsilon }\hfill \\ -\xi {\int }_{\Omega }{u}_{\epsilon }^{\beta -1}\nabla u_{\epsilon }·\nabla w_{\epsilon }+\kappa |\Omega |+\kappa {\int }_{\Omega }ln{u}_{\epsilon }\hfill \\ -\mu {\int }_{\Omega }{u}_{\epsilon }^m-\mu {\int }_{\Omega }{u}_{\epsilon }^{m}ln{u}_{\epsilon }\hfill \\ \le \chi {\int }_{\Omega }{u}_{\epsilon }^{\alpha -1}\nabla u_{\epsilon }·\nabla v_{\epsilon }-\xi {\int }_{\Omega }{u}_{\epsilon }^{\beta -1}\nabla u_{\epsilon }·\nabla w_{\epsilon }-{\displaystyle \frac{\mu }{2}}{\int }_{\Omega }{u}_{\epsilon }^m+C_4,\hfill \end{array}$$

(3.5)

which used ${\int }_{\Omega }\ln u_{\epsilon }\le {\int }_{\Omega }{u}_{\epsilon }$ and $-u_{\epsilon }\ln u_{\epsilon }\le e^{-1}$.

Multiplying ${(2.1)}_2$ by $-▵v_{\epsilon }$, integrating by part, we have

$$\begin{array}{c}{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }|\nabla v_{\epsilon }{|}^2=-{\int }_{\Omega }|▵v_{\epsilon }{|}^2-\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2-\eta {\int }_{\Omega }{u}_{\epsilon }^{k_1}▵v_{\epsilon }\hfill \\ \le -{\int }_{\Omega }|▵v_{\epsilon }{|}^2-\sigma {\int }_{\Omega }|\nabla v_{\epsilon }{|}^2+{\eta }^2{\int }_{\Omega }{u}_{\epsilon }^{2k_1}+{\displaystyle \frac{1}{4}}{\int }_{\Omega }{|▵v_{\epsilon }|}^2\hfill \\ \le -{\displaystyle \frac{3}{4}}{\int }_{\Omega }|▵v_{\epsilon }{|}^2-\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2+{\eta }^2{\int }_{\Omega }{u}_{\epsilon }^{2k_1}\hfill \end{array}$$

(3.6)

for all $t\in (0,T_{max})$. According to the Young’s inequality, we estimate

$$\begin{array}{c}\chi {\int }_{\Omega }{u}_{\epsilon }^{\alpha -1}\nabla u_{\epsilon }·\nabla v_{\epsilon }=-{\displaystyle \frac{\chi }{\alpha }}{\int }_{\Omega }{u}_{\epsilon }^{\alpha }▵v_{\epsilon }\le {\displaystyle \frac{{\chi }^2}{{\alpha }^2}}{\int }_{\Omega }{u}_{\epsilon }^{2\alpha }+{\displaystyle \frac{1}{4}}{\int }_{\Omega }{|▵v_{\epsilon }|}^2,\hfill \end{array}$$

(3.7)

as well as

$$\begin{array}{c}-\xi {\int }_{\Omega }{u}_{\epsilon }^{\beta -1}\nabla u_{\epsilon }·\nabla w_{\epsilon }={\displaystyle \frac{\xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{\beta }▵w_{\epsilon }\hfill \\ ={\displaystyle \frac{\delta \xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{\beta }{w}_{\epsilon }-{\displaystyle \frac{\gamma \xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{k_2+\beta }\hfill \\ \le {\int }_{\Omega }{u}_{\epsilon }^{2\beta }+{\displaystyle \frac{{\delta }^2{\xi }^2}{4{\beta }^2}}{\int }_{\Omega }{w}_{\epsilon }^2-{\displaystyle \frac{\gamma \xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{k_2+\beta }\hfill \\ \le {\int }_{\Omega }{u}_{\epsilon }^{2\beta }+\sigma {\int }_{\Omega }{u}^2-{\displaystyle \frac{\gamma \xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{k_2+\beta }+C_5\hfill \end{array}$$

(3.8)

which used Lemma 2.3 and for all $t\in (0,T_{max})$. Since $2\sigma {\int }_{\Omega }{u}_{\epsilon }ln{u}_{\epsilon }\le 2\sigma {\int }_{\Omega }{u}_{\epsilon }^2$, and collecting (3.5)–(3.8),we have

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }(u_{\epsilon }ln{u}_{\epsilon }+{\displaystyle \frac{1}{2}}|\nabla v_{\epsilon }{|}^2)+2\sigma {\int }_{\Omega }{u}_{\epsilon }ln{u}_{\epsilon }+\sigma {\int }_{\Omega }{|\nabla v_{\epsilon }|}^2\hfill \\ \le 3\sigma {\int }_{\Omega }{u}_{\epsilon }^2+{\displaystyle \frac{{\chi }^2}{{\alpha }^2}}{\int }_{\Omega }{u}_{\epsilon }^{2\alpha }+{\int }_{\Omega }{u}_{\epsilon }^{2\beta }+{\eta }^2{\int }_{\Omega }{u}_{\epsilon }^{2k_1}-{\displaystyle \frac{\gamma \xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{k_2+\beta }-{\displaystyle \frac{\mu }{2}}{\int }_{\Omega }{u}_{\epsilon }^m+C_6.\hfill \end{array}$$

(3.9)

It follows from $k_2>2k_1-\beta$, $m>max\{2,2\alpha ,2\beta \}$ and Young’s inequality that

$$\begin{array}{c}3\sigma {\int }_{\Omega }{u}_{\epsilon }^2+{\displaystyle \frac{{\chi }^2}{{\alpha }^2}}{\int }_{\Omega }{u}_{\epsilon }^{2\alpha }+{\int }_{\Omega }{u}_{\epsilon }^{2\beta }+{\eta }^2{\int }_{\Omega }{u}_{\epsilon }^{2k_1}-{\displaystyle \frac{\gamma \xi }{\beta }}{\int }_{\Omega }{u}_{\epsilon }^{k_2+\beta }-{\displaystyle \frac{\mu }{2}}{\int }_{\Omega }{u}_{\epsilon }^m\le C_7.\hfill \end{array}$$

(3.10)

Let $y(t)={\int }_{\Omega }(u_{\epsilon }ln{u}_{\epsilon }+{\displaystyle \frac{1}{2}}|\nabla v_{\epsilon }{|}^2)$, together with (3.9) and (3.10), this indicates

$$\begin{array}{c}{y}^{\prime }(t)+2\sigma y(t)\le C_8.\hfill \end{array}$$

(3.11)

So we have for all $t\in (0,T_{max})$ that

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon }ln{u}_{\epsilon }+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{|\nabla v_{\epsilon }|}^2\le C_9.\hfill \end{array}$$

(3.12)

Then we derive

$$\begin{array}{c}{\displaystyle \frac{1}{2}}{\int }_{\Omega }|\nabla v_{\epsilon }{|}^2\le -{\int }_{\Omega }{u}_{\epsilon }ln{u}_{\epsilon }+C_9\le e^{-1}|\Omega |+C_9\hfill \end{array}(3.13)$$

(3.13)

for all $t\in (0,T_{max})$ and the desired results are proved.□

Lemma 3.2.

Let $p\ge 2$ and assume that (1.2)(1.3) hold with $m>1,k_1>0$ and $0<k_2\le 1$. If $m>max\{2k_1,{\displaystyle \frac{p\alpha }{p-1}}-1\}$ or $k_2>max\{2k_1-\beta ,{\displaystyle \frac{p\alpha }{p-1}}-\beta -1\}$ with $m>max\{2,2\alpha ,2\beta \}$, then for any $q>1$, one can find a constant $C_{10}>0$ such that for any $t\in (0,T_{max})$

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon }^q(x,t)dx\le C_{10}\hfill \end{array}$$

(3.14)

for any $\epsilon \in (0,1)$.

Proof. 

Multiply ${(2.1)}_1$ by $q{u}_{\epsilon }^{q-1}$ integrate over $\Omega$ ,we can find that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{u}_{\epsilon }^q+q(q-1){\int }_{\Omega }{u}_{\epsilon }^{q-2}(|\nabla u_{\epsilon }{|}^2{+\epsilon )}^{{\displaystyle \frac{p-2}{2}}}{|\nabla u_{\epsilon }|}^2\hfill \\ \le \chi q(q-1){\int }_{\Omega }{u}_{\epsilon }^{q+\alpha -2}\nabla u_{\epsilon }·\nabla v_{\epsilon }+{\displaystyle \frac{\xi q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta -1}▵w_{\epsilon }\hfill \\ +\kappa q{\int }_{\Omega }{u}_{\epsilon }^{q-1}-\mu q{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}\hfill \end{array}$$

(3.15)

for all $t\in (0,T_{max})$. Note that for $p\ge 2$ and $t\in (0,T_{max})$,

$$\begin{array}{c}(|\nabla u_{\epsilon }{|}^2{+\epsilon )}^{{\displaystyle \frac{p-2}{2}}}|\nabla u_{\epsilon }{|}^2\ge {|\nabla u_{\epsilon }|}^p.\hfill \end{array}$$

(3.16)

Together with (3.15) (3.16), and Young’s inequality, it follows that for $p\ge 2$

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{u}_{\epsilon }^q+q(q-1){\int }_{\Omega }{u}_{\epsilon }^{q-2}{|\nabla u_{\epsilon }|}^p\hfill \\ \le \chi q(q-1){\int }_{\Omega }{u}_{\epsilon }^{q+\alpha -2}\nabla u_{\epsilon }·\nabla v_{\epsilon }+{\displaystyle \frac{\xi q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta -1}▵w_{\epsilon }\hfill \\ +\kappa q{\int }_{\Omega }{u}_{\epsilon }^{q-1}-\mu q{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}\hfill \\ \le \chi q(q-1){\int }_{\Omega }{u}_{\epsilon }^{q+\alpha -2}\nabla u_{\epsilon }·\nabla v_{\epsilon }+{\displaystyle \frac{\xi q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta -1}▵w_{\epsilon }\hfill \\ -{\displaystyle \frac{\mu q}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}+C_{11}\hfill \end{array}(3.17)$$

(3.17)

for all $t\in (0,T_{max})$, where $C_{11}$ is a positive constant. As a consequence of Young’s inequality, we have

$$\begin{array}{c}\chi q(q-1){\int }_{\Omega }{u}_{\epsilon }^{q+\alpha -2}\nabla u_{\epsilon }·\nabla v_{\epsilon }\hfill \\ \le {\displaystyle \frac{q(q-1)}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q-2}|\nabla u_{\epsilon }{|}^p+C_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}{|\nabla v_{\epsilon }|}^{{\displaystyle \frac{p}{p-1}}},\hfill \end{array}$$

(3.18)

where $C_{12}$ is a positive constant. Quoting Lemma 2.3 and applying Young’s inequality to get constants $C_{13},C_{14}>0$ such that

$$\begin{array}{c}{\displaystyle \frac{\xi q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta -1}▵w_{\epsilon }\hfill \\ ={\displaystyle \frac{\xi \delta q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta -1}{w}_{\epsilon }-{\displaystyle \frac{\xi \gamma q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}\hfill \\ \le {\displaystyle \frac{\xi \gamma q(q-1)}{4(q+\beta -1)}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}+C_{13}{\int }_{\Omega }{w}_{\epsilon }^{{\displaystyle \frac{q+\beta +k_2-1}{k_2}}}-{\displaystyle \frac{\xi \gamma q(q-1)}{q+\beta -1}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}\hfill \\ =C_{13}{\int }_{\Omega }{w}_{\epsilon }^{{\displaystyle \frac{q+\beta +k_2-1}{k_2}}}-{\displaystyle \frac{3\xi \gamma q(q-1)}{4(q+\beta -1)}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}\hfill \\ \le -{\displaystyle \frac{\xi \gamma q(q-1)}{2(q+\beta -1)}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}+C_{14}\hfill \end{array}$$

(3.19)

for all $t\in (0,T_{max})$. Together with (3.17) (3.18) and (3.19), we have

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{u}_{\epsilon }^q+{\displaystyle \frac{q(q-1)}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q-2}{|\nabla u_{\epsilon }|}^p\hfill \\ \le C_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}{|\nabla v_{\epsilon }|}^{{\displaystyle \frac{p}{p-1}}}-{\displaystyle \frac{\xi \gamma q(q-1)}{2(q+\beta -1)}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}\hfill \\ -{\displaystyle \frac{\mu q}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}+C_{11}+C_{14}\hfill \end{array}$$

(3.20)

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

Due to the fact that

$$\begin{array}{c}{\displaystyle \frac{q(q-1)}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q-2}|\nabla u_{\epsilon }{|}^p={\displaystyle \frac{q(q-1)p^p}{2{(p+1-2)}^p}}{\parallel \nabla u_{\epsilon }^{{\displaystyle \frac{p+q-2}{p}}}\parallel }_{L^p(\Omega )}^p\hfill \end{array}$$

(3.21)

for all $t\in (0,T_{max})$. From (3.20) and (3.21), we conclude that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }{u}_{\epsilon }^q+{\displaystyle \frac{q(q-1)p^p}{2{(p+1-2)}^p}}{\parallel \nabla u_{\epsilon }^{{\displaystyle \frac{p+q-2}{p}}}\parallel }_{L^p(\Omega )}^p\hfill \\ \le C_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}{|\nabla v_{\epsilon }|}^{{\displaystyle \frac{p}{p-1}}}-{\displaystyle \frac{\xi \gamma q(q-1)}{2(q+\beta -1)}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}\hfill \\ -{\displaystyle \frac{\mu q}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}+C_{11}+C_{14}\hfill \end{array}$$

(3.22)

for all $t\in (0,T_{max})$. Analogous to Lemma 3.11 in [25],we can choose a positive constant $C_{15}$ for any $q^{\prime }\ge 2$ such that the following inequality holds

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\int }_{\Omega }|\nabla v_{\epsilon }{|}^{2q^{\prime }}+{\displaystyle \frac{2(q^{\prime }-1)}{q^{\prime }}}{\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{|}^2\le C_{15}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}{|\nabla v_{\epsilon }|}^{2q^{\prime }-2}+C_{15}.\hfill \end{array}$$

(3.23)

From the combination of (3.22) and (3.23), it follows that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}({\int }_{\Omega }{u}_{\epsilon }^q+{\int }_{\Omega }|{\nabla }_{\epsilon }{|}^{2q^{\prime }})+{\displaystyle \frac{q(q-1)p^p}{2{(p+1-2)}^p}}\parallel \nabla u_{\epsilon }^{{\displaystyle \frac{p+q-2}{p}}}{\parallel }_{L^p(\Omega )}^p+{\displaystyle \frac{2(q^{\prime }-1)}{q^{\prime }}}{\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{{|}^{q^{\prime }}|}^2\hfill \\ \le C_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}|\nabla v_{\epsilon }{|}^{{\displaystyle \frac{p}{p-1}}}+C_{15}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}{|\nabla v_{\epsilon }|}^{2q^{\prime }-2}\hfill \\ -{\displaystyle \frac{\xi \gamma q(q-1)}{2(q+\beta -1)}}{\int }_{\Omega }{u}_{\epsilon }^{q+\beta +k_2-1}-{\displaystyle \frac{\mu q}{2}}{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}+C_{16}\hfill \end{array}$$

(3.24)

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

Case (i): $m>max\{2k_1,{\displaystyle \frac{p\alpha }{p-1}}-1\}$. Let $q^{\prime }:={\displaystyle \frac{(p-1)(q+m-1)}{(p-1)(m+1)-p\alpha }}$ and take $q\ge m+3-{\displaystyle \frac{2p\alpha }{p-1}}$,and $m>{\displaystyle \frac{p\alpha }{p-1}}-1$ , we have $q^{\prime }\ge 2$. Useing H$\ddot{\mathrm{o}}$lder inequality, we obtain

$$\begin{array}{c}{C}_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}|\nabla v_{\epsilon }{|}^{{\displaystyle \frac{p}{p-1}}}\le C_{12}{({\int }_{\Omega }{u}_{\epsilon }^{q+m-1})}^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{(p-1)(q+m-1)}}}({\int }_{\Omega }|\nabla v_{\epsilon }{{|}^{{\displaystyle \frac{p{q}^{\prime }}{p-1}}})}^{{\displaystyle \frac{1}{q^{\prime }}}}\hfill \end{array}$$

(3.25)

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

$$\begin{array}{c}{C}_{15}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}|\nabla v_{\epsilon }{|}^{2q^{\prime }-2}\le C_{15}{({\int }_{\Omega }{u}_{\epsilon }^{q+m-1})}^{{\displaystyle \frac{2k_1}{q+m-1}}}({\int }_{\Omega }|\nabla v_{\epsilon }{{|}^{(2q^{\prime }-2)\lambda })}^{{\displaystyle \frac{1}{\lambda }}},\hfill \end{array}$$

(3.26)

where $\lambda ={\displaystyle \frac{q+m-1}{q+m-2k_1-1}}>1$. Then by means of the Gagliardo–Nirenberg inequality, we deduce

$$\begin{array}{c}({\int }_{\Omega }|\nabla v_{\epsilon }{|}^{{\displaystyle \frac{p{q}^{\prime }}{p-1}}}{)}^{{\displaystyle \frac{1}{q^{\prime }}}}=\parallel |\nabla v_{\epsilon }{{|}^{q^{\prime }}\parallel }_{L^{{\displaystyle \frac{p}{p-1}}}(\Omega )}^{{\displaystyle \frac{p}{(p-1)q^{\prime }}}}\hfill \\ \le (C_{17}\parallel \nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^2(\Omega )}^{{\theta }_1}\parallel |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^{{\displaystyle \frac{2}{q^{\prime }}}}(\Omega )}^{1-{\theta }_1}+C_{17}\parallel |\nabla v_{\epsilon }{|}^{q^{\prime }}{{\parallel }_{L^{{\displaystyle \frac{2}{q^{\prime }}}}(\Omega )})}^{{\displaystyle \frac{p}{(p-1)q^{\prime }}}}\hfill \end{array}$$

(3.27)

for all $t\in (0,T_{max})$, where ${\theta }_1={\displaystyle \frac{\frac{q^{\prime }}{2}-\frac{p-1}{p}}{\frac{q^{\prime }}{2}+\frac{1}{n}-\frac{1}{2}}}$. Since $p\ge 2$, selecting $q>{\displaystyle \frac{2[(p-1)(m+1)-p\alpha ]}{p}}-m+1$, we have ${\theta }_1\in (0,1)$. We invoke (3.27) and Lemma 3.1 with $m>2k_1$ such that

$$\begin{array}{c}({\int }_{\Omega }|\nabla v_{\epsilon }{|}^{{\displaystyle \frac{p{q}^{\prime }}{p-1}}}{)}^{{\displaystyle \frac{1}{q^{\prime }}}}\le C_{18}({\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{{|}^2)}^{{\displaystyle \frac{p{\theta }_1}{2(p-1)q^{\prime }}}}+C_{18}\hfill \end{array}(3.28)$$

(3.28)

with $C_{18}>0$. Recalling the Gagliardo–Nirenberg inequality again we estimate

$$\begin{array}{c}({\int }_{\Omega }|\nabla v_{\epsilon }{|}^{2(q^{\prime }-1)\lambda }{)}^{{\displaystyle \frac{1}{\lambda }}}=\parallel |\nabla v_{\epsilon }{{|}^{q^{\prime }}\parallel }_{L^{{\displaystyle \frac{2\lambda (q^{\prime }-1)}{q^{\prime }}}}(\Omega )}^{{\displaystyle \frac{2(q^{\prime }-1)}{q^{\prime }}}}\hfill \\ \le (C_{19}\parallel \nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^2(\Omega )}^{{\theta }_2}\parallel |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^{{\displaystyle \frac{2}{q^{\prime }}}}(\Omega )}^{1-{\theta }_2}+C_{19}\parallel |\nabla v_{\epsilon }{|}^{q^{\prime }}{{\parallel }_{L^{{\displaystyle \frac{2}{q^{\prime }}}}(\Omega )})}^{{\displaystyle \frac{2(q^{\prime }-1)}{q^{\prime }}}}\hfill \end{array}$$

(3.29)

with $C_{19}>0$, where ${\theta }_2={\displaystyle \frac{\frac{q^{\prime }}{2}-\frac{q^{\prime }}{2\lambda (q^{\prime }-1)}}{\frac{q^{\prime }}{2}+\frac{1}{n}-\frac{1}{2}}}$. Due to $m>max\{1,2k_1\},p\ge 2$ and take $q>max\{m+3-{\displaystyle \frac{2p\alpha }{p-1}},{\displaystyle \frac{(n-2)p\alpha }{2(p-1)}}-{\displaystyle \frac{n(m-2k_1)}{2}}+2-{\displaystyle \frac{n}{2}}\}$, we get ${\theta }_2\in (0,1)$. Then combining Lemma 3.1, we have

$$\begin{array}{c}({\int }_{\Omega }|\nabla v_{\epsilon }{|}^{2(q^{\prime }-1)\lambda }{)}^{{\displaystyle \frac{1}{\lambda }}}\le C_{20}({\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{{|}^2)}^{{\displaystyle \frac{(q^{\prime }-1){\theta }_2}{q^{\prime }}}}+C_{20}.\hfill \end{array}$$

(3.30)

Together with (3.25),(3.26), (3.28) and (3.30), we obtain

$$\begin{array}{c}{C}_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}|\nabla v_{\epsilon }{|}^{{\displaystyle \frac{p}{p-1}}}+C_{15}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}{|\nabla v_{\epsilon }|}^{2q^{\prime }-2}\hfill \\ \le C_{12}{C}_{18}{({\int }_{\Omega }{u}_{\epsilon }^{q+m-1})}^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{(p-1)(q+m-1)}}}({\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{{|}^2)}^{{\displaystyle \frac{p{\theta }_1}{2(p-1)q^{\prime }}}}\hfill \\ +C_{12}{C}_{18}{({\int }_{\Omega }{u}_{\epsilon }^{q+m-1})}^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{(p-1)(q+m-1)}}}\hfill \\ +C_{15}{C}_{20}{({\int }_{\Omega }{u}_{\epsilon }^{q+m-1})}^{{\displaystyle \frac{2k_1}{q+m-1}}}({\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{{|}^2)}^{{\displaystyle \frac{(q^{\prime }-1){\theta }_2}{q^{\prime }}}}\hfill \\ +C_{15}{C}_{20}{({\int }_{\Omega }{u}_{\epsilon }^{q+m-1})}^{{\displaystyle \frac{2k_1}{q+m-1}}}\hfill \end{array}$$

(3.31)

for $p\ge 2$ and $q>max\{m+3-{\displaystyle \frac{2p\alpha }{p-1}},{\displaystyle \frac{(n-2)p\alpha }{2(p-1)}}-{\displaystyle \frac{n(m-2k_1)}{2}}+2-{\displaystyle \frac{n}{2}},{\displaystyle \frac{2[(p-1)(m+1)-p\alpha ]}{p}}-m+1\}$.

Next we prove ${\displaystyle \frac{(p-1)(q-1)+1}{(p-1)(q+m-1)}}+{\displaystyle \frac{p{\theta }_1}{2(p-1)q^{\prime }}}<1$.This is an obvious fact that holds true,otherwise, if ${\displaystyle \frac{(p-1)(q-1)+1}{(p-1)(q+m-1)}}+{\displaystyle \frac{p{\theta }_1}{2(p-1)q^{\prime }}}\ge 1$,this leads to

$$\begin{array}{c}{\theta }_1\ge {\displaystyle \frac{2(p-1)q^{\prime }}{p}}·{\displaystyle \frac{(p-1)(m+1)-p\alpha }{(p-1)(q+m-1)}}.\hfill \\ =2-{\displaystyle \frac{2}{p}}\hfill \\ \ge 1\hfill \end{array}$$

(3.32)

which used $q^{\prime }={\displaystyle \frac{(p-1)(q+m-1)}{(p-1)(m+1)-p\alpha }}$ and $p\ge 2$. This means that

$$\begin{array}{c}{\displaystyle \frac{(p-1)(q-1)+1}{(p-1)(q+m-1)}}+{\displaystyle \frac{p{\theta }_1}{2(p-1)q^{\prime }}}<1.\hfill \end{array}$$

(3.33)

Subsequently,we prove ${\displaystyle \frac{2k_1}{q+m-1}}+{\displaystyle \frac{(q^{\prime }-1){\theta }_2}{q^{\prime }}}<1$. Let

$$\begin{array}{c}{\displaystyle \frac{2k_1}{q+m-1}}+{\displaystyle \frac{(q^{\prime }-1){\theta }_2}{q^{\prime }}}={\displaystyle \frac{h_1(p,q)}{h_2(p,q)}}.\hfill \end{array}$$

(3.34)

where

$$\begin{array}{c}{h}_1(p,q)=2k_1[(p-1)(q+m-1)+{\displaystyle \frac{2-2n}{n}}((p-1)(m+1)-p\alpha )]\hfill \\ -(q+m-1)((p-1)(q-2)+p\alpha )\hfill \\ +(q+m-2k_1-1)((p-1)(q-2)+p\alpha )\hfill \end{array}$$

(3.35)

and

$$\begin{array}{c}{h}_2(p,q)=(q+m-1)[(p-1)(q+m-1)+{\displaystyle \frac{2-2n}{n}}((p-1)(m+1)-p\alpha )].\hfill \end{array}$$

(3.36)

A direct calculation yields

$$\begin{array}{c}{h}_2(p,q)-h_1(p,q)=(q+m-2k_1-1)[(p-1)(q+m-1)\hfill \\ +{\displaystyle \frac{2-2n}{n}}((p-1)(m+1)-p\alpha )]\hfill \\ +(q+m-1)((p-1)(q-2)+p\alpha ),\hfill \end{array}$$

(3.37)

Due to $q^{\prime }\ge 2$,we have $(p-1)(q+m-1)\ge 2[(p-1)(m+1)-p\alpha ]$.Combining $m>max\{2k_1,{\displaystyle \frac{p\alpha }{p-1}}-1\}$, $p\ge 2$ and $q>1$ ,we have

$$\begin{array}{c}{h}_2(p,q)-h_1(p,q)\ge {\displaystyle \frac{2}{n}}(q+m-2k_1-1)((p-1)(m+1)-p\alpha )\hfill \\ +(q+m-1)((p-1)(q-2)+p\alpha )\hfill \\ >0.\hfill \end{array}$$

(3.38)

It means

$$\begin{array}{c}{\displaystyle \frac{2k_1}{q+m-1}}+{\displaystyle \frac{(q^{\prime }-1){\theta }_2}{q^{\prime }}}<1.\hfill \end{array}$$

(3.39)

Together with (3.31), (3.33), (3.39) and by the Young’s inequality twice, we deduce

$$\begin{array}{c}{C}_{12}{\int }_{\Omega }{u}_{\epsilon }^{{\displaystyle \frac{(p-1)(q-2)+p\alpha }{p-1}}}|\nabla v_{\epsilon }{|}^{{\displaystyle \frac{p}{p-1}}}+C_{15}{\int }_{\Omega }{u}_{\epsilon }^{2k_1}{|\nabla v_{\epsilon }|}^{2q^{\prime }-2}\hfill \\ \le ϵ({\int }_{\Omega }{u}_{\epsilon }^{q+m-1}+{\int }_{\Omega }|\nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{|}^2)+C_{21}\hfill \end{array}(3.40)$$

(3.40)

for any $ϵ>0$, $p\ge 2$ and $q>q_0:=max\{1,m+3-{\displaystyle \frac{2p\alpha }{p-1}},{\displaystyle \frac{n-2}{2(p-1)}}-{\displaystyle \frac{n(m-2k_1)}{2}}+1,{\displaystyle \frac{2[(p-1)m-1]}{p}}-m+1\}$. Using the Gagliardo–Nirenberg inequality, there exists $C_{22}>0$ such that

$$\begin{array}{c}{\int }_{\Omega }|\nabla v_{\epsilon }{|}^{2q^{\prime }}=\parallel |\nabla v_{\epsilon }{{|}^{q^{\prime }}\parallel }_{L^2(\Omega )}^2\hfill \\ \le C_{22}\parallel \nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^2(\Omega )}^{2{\theta }_3}\parallel |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^{{\displaystyle \frac{2}{q^{\prime }}}}(\Omega )}^{2(1-{\theta }_3)}+C_{22}\parallel |\nabla v_{\epsilon }{{|}^{q^{\prime }}\parallel }_{L^{{\displaystyle \frac{2}{q^{\prime }}}}(\Omega )},\hfill \end{array}$$

(3.41)

where ${\theta }_3={\displaystyle \frac{\frac{q^{\prime }}{2}-\frac{1}{2}}{\frac{q^{\prime }}{2}+\frac{1}{n}-\frac{1}{2}}}\in (0,1)$. From (3.41) and Lemma 3.1, we conclude taht

$$\begin{array}{c}{\int }_{\Omega }|\nabla v_{\epsilon }{|}^{2q^{\prime }}\le C_{23}\parallel \nabla |\nabla v_{\epsilon }{|}^{q^{\prime }}{\parallel }_{L^2(\Omega )}^2\parallel +C_{23}\hfill \end{array}$$

(3.42)

Using Young’s inequality again, we obtain

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon }^q\le C_{24}{\int }_{\Omega }{u}_{\epsilon }^{q+m-1}+C_{24}\hfill \end{array}$$

(3.43)

for all $t\in (0,T_{max})$. Combine (3.24), (3.40), (3.42) and (3.43), we can see that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}({\int }_{\Omega }{u}_{\epsilon }^q+{\int }_{\Omega }|{\nabla }_{\epsilon }{|}^{2q^{\prime }})+{\int }_{\Omega }{u}_{\epsilon }^q+{\int }_{\Omega }{|{\nabla }_{\epsilon }|}^{2q^{\prime }}\le C_{25}\hfill \end{array}$$

(3.44)

for all $q>q_0>1$. By the Young’s inequality again, we obtain for $q>1$ that

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon }^q\le C_{26}\hfill \end{array}$$

(3.45)

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

Case (ii): $k_2>max\{2k_1-\beta ,{\displaystyle \frac{p\alpha }{p-1}}-\beta -1\}$ with $m>max\{2,2\alpha ,2\beta \}$. Following the approach used for Case (i), we reach corresponding outcomes.□

Lemma 3.3.

Under assumption of Lemma 3.2, there exists $C_{27}>0$ such that for any $\epsilon \in (0,1)$

$$\begin{array}{c}\parallel v_{\epsilon }{(·,t)\parallel }_{W^{1,\infty }(\Omega )}\le C_{27},{\parallel w_{\epsilon }(·,t)\parallel }_{W^{1,\infty }(\Omega )}\le C_{27}\hfill \end{array}$$

(3.46)

and

$$\begin{array}{c}\parallel u_{\epsilon }{(·,t)\parallel }_{L^{\infty }(\Omega )}\le C_{27}\hfill \end{array}(3.47)$$

(3.47)

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

Proof. 

In analogy with Lemma 3.3 of [32] , we skip the proof.□

4. Proof of Theorem 1.1.

The existence of global weak solutions to system (1.1) can be shown by taking limits of $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$. We start this analysis with a fundamental lemma from [12,16].

Lemma 4.1.

Under assumption of Lemma 3.2, there exist constants $C_1,C_2>0$ such that for all $t>0$

$$\begin{array}{c}{\int }_0^t{\int }_{\Omega }{|\nabla u_{\epsilon }(·,t)|}^p\le C_1\hfill \end{array}$$

(4.1)

and for all $T>0$

$$\begin{array}{c}\parallel {\partial }_t{u}_{\epsilon }{\parallel }_{L^1((0,T);{(W_0^{2,p}(\Omega ))}^{*})}\le C_2.\hfill \end{array}$$

(4.2)

Proof. 

Lemma 3.2’s proof establishes that ${\int }_0^t{\int }_{\Omega }{u}_{\epsilon }^{q-2}{|\nabla u_{\epsilon }|}^p\le C_3$. Let $q=2$, we deduce (4.1).

Let $\psi \in C_0^{\infty }(\Omega )$ satisfies ${\parallel \psi \parallel }_{W^{2,p}(\Omega )}\le 1$. Multiplying ${(2.1)}_1$ by $\psi$, integrating by part, we obtain

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon t}\psi \le -{\int }_{\Omega }(|\nabla u_{\epsilon }{{|}^2+\epsilon )}^{{\displaystyle \frac{p-2}{2}}}\nabla u_{\epsilon }·\nabla \psi +\chi {\int }_{\Omega }{u}_{\epsilon }^{\alpha }\nabla v_{\epsilon }·\nabla \psi \hfill \\ -\xi {\int }_{\Omega }{u}_{\epsilon }^{\beta }\nabla w_{\epsilon }·\nabla \psi +\kappa {\int }_{\Omega }\psi -\mu {\int }_{\Omega }{u}_{\epsilon }^m\psi .\hfill \end{array}$$

(4.3)

Since $p\ge 2$, using the H$\ddot{\mathrm{o}}$lder inequality and Young’s inequality, we can obtain

$$\begin{array}{c}|{\int }_{\Omega }(|\nabla u_{\epsilon }{|}^2+\epsilon {)}^{{\displaystyle \frac{p-2}{2}}}\nabla u_{\epsilon }·\nabla \psi |\hfill \\ \le C_4{\int }_{\Omega }(|\nabla u_{\epsilon }{|}^{p-1}+1)\nabla |\psi |\hfill \\ \le C_5({\int }_{\Omega }|\nabla u_{\epsilon }{|}^p{)}^{{\displaystyle \frac{p-1}{p}}}({\int }_{\Omega }{|\nabla \psi |}^p{)}^{{\displaystyle \frac{1}{p}}}+C_6({\int }_{\Omega }{|\nabla \psi |}^p{)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \le C_7{\int }_{\Omega }{|\nabla u_{\epsilon }|}^p+C_8\hfill \end{array}$$

(4.4)

with $C_i>0(i=4,5,6,7,8)$. Due to Lemma 3.3, we can find constants $C_9,C_{10}>0$ such that

$$\begin{array}{c}|\chi {\int }_{\Omega }{u}_{\epsilon }^{\alpha }\nabla v_{\epsilon }·\nabla \psi |\le \chi \parallel u_{\epsilon }{\parallel }_{L^{\infty }(\Omega )}^{\alpha }\parallel \nabla v_{\epsilon }{\parallel }_{L^{{\displaystyle \frac{p}{p-1}}}(\Omega )}{\parallel \nabla \psi \parallel }_{L^p(\Omega )}\le C_9\hfill \end{array}$$

(4.5)

and

$$\begin{array}{c}|\xi {\int }_{\Omega }{u}_{\epsilon }^{\beta }\nabla w_{\epsilon }·\nabla \psi |\le \xi \parallel u_{\epsilon }{\parallel }_{L^{\infty }(\Omega )}^{\beta }\parallel \nabla w_{\epsilon }{\parallel }_{L^{{\displaystyle \frac{p}{p-1}}}(\Omega )}{\parallel \nabla \psi \parallel }_{L^p(\Omega )}\le C_{10}.\hfill \end{array}$$

(4.6)

Similarly, we can get

$$\begin{array}{c}|\kappa {\int }_{\Omega }\psi -\mu {\int }_{\Omega }{u}_{\epsilon }^l\psi |\le C_{11}{\parallel \psi \parallel }_{L^1(\Omega )}(1+\parallel u_{\epsilon }{\parallel }_{L^{\infty }(\Omega )}^m)\le C_{12}\hfill \end{array}(4.7)$$

with $C_{11},C_{12}>0$. Combining (4.3)-(4.7) yields

$$\begin{array}{c}{\int }_{\Omega }{u}_{\epsilon t}\psi \le C_7{\int }_{\Omega }{|\nabla u_{\epsilon }|}^p+C_{13}\hfill \end{array}$$

(4.8)

Integrating (4.8) on $(0,T)$ and collecting (4.1) yields (4.2).□

Lemma 4.1.

let $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$ be the classical solution of (2.1). There exists a nonnegative funtion $(u,v,w)$ such that as $\epsilon \to 0$,

$$\begin{array}{c}{u}_{\epsilon }\to u\mathrm{a}.\mathrm{e}.\mathrm{in}{Q}_T,\hfill \end{array}$$

(4.9)

$$\begin{array}{c}{u}_{\epsilon }\to u\mathrm{in}{L}_{loc}^p(Q_T),\hfill \end{array}$$

(4.10)

$$\begin{array}{c}{u}_{\epsilon }{\rightharpoonup }^{*}u\mathrm{in}{L}^{\infty }(Q_T),\hfill \end{array}$$

(4.11)

$$\begin{array}{c}\nabla u_{\epsilon }\rightharpoonup \nabla u\mathrm{in}{L}_{loc}^p(Q_T),\hfill \end{array}$$

(4.12)

$$\begin{array}{c}(|\nabla u_{\epsilon }{|}^2{+\epsilon )}^{{\displaystyle \frac{p-2}{2}}}\nabla u_{\epsilon }\rightharpoonup {|\nabla u|}^{p-2}\nabla u\mathrm{in}{L}_{loc}^{{\displaystyle \frac{p}{p-1}}}(Q_T),\hfill \end{array}$$

(4.13)

$$\begin{array}{c}{v}_{\epsilon }\to v,uniformly,\hfill \end{array}$$

(4.14)

$$\begin{array}{c}{v}_{\epsilon }\rightharpoonup v\mathrm{in}{W}_r^{2,1}(Q_T),foranyr>1,\hfill \end{array}$$

(4.15)

$$\begin{array}{c}{w}_{\epsilon }\rightharpoonup w,in{L}_{loc}^2((0,T),W^{1,2}(\Omega )),\hfill \end{array}$$

(4.16)

$$\begin{array}{c}{w}_{\epsilon }{\rightharpoonup }^{*}w,in{L}^{\infty }(\Omega \times (0,T)),\hfill \end{array}$$

(4.17)

$$\begin{array}{c}▽w_{\epsilon }{\rightharpoonup }^{*}▽w,in{L}^{\infty }(\Omega \times (0,T)),\hfill \end{array}$$

(4.18)

for any $T>0$.

Proof. 

Through Lemmas 3.3 and 4.1 and together with Aubin-Lions lemma, we obtain a nonnegative function u satisfies as $\epsilon \to 0$,

$$\begin{array}{c}{u}_{\epsilon }\to u\mathrm{in}{L}_{loc}^p(Q_T)\mathrm{and}\mathrm{a}.\mathrm{e}.\mathrm{in}{Q}_T,\hfill \end{array}$$

(4.19)

this yields (4.9) and (4.10).

Collecting Lemma 3.3 and (4.19),we deduce (4.11). Together with (4.1) and (4.19) that (4.12).

Collecting (4.11), (4.19) and Lemma 3.3, we obtain (4.13) by the method of the Lemma 4.5 in [28]. We may further conclude from Lemma 3.3 in conjunction with (1.3) that

$$\begin{array}{c}\underset{t\in (1,\infty )}{sup}{\int }_{t-1}^t{\parallel g_1(u_{\epsilon })\parallel }_{L^r}^{r}ds\le C_{14}\hfill \end{array}$$

(4.20)

for any $r>1$. Collecting Lemma 2.4 in [5] and (4.20), we have

$$\begin{array}{c}\underset{t\in (1,\infty )}{sup}{\int }_{t-1}^t(\parallel v_{\epsilon }{\parallel }_{W^{2,r}}^r+\parallel v_{\epsilon t}{\parallel }_{L^r}^r)ds\le C_{15},\hfill \end{array}$$

(4.21)

this implies

$$\begin{array}{c}\parallel v_{\epsilon }{\parallel }_{W_r^{2,1}(\Omega \times (t-1,t))}\le C_{16}\hfill \end{array}$$

(4.22)

with $C_{16}>0$. In view of (4.22), we obtain (4.15). Since $W_r^{2,1}(Q_T)\hookrightarrow C^{2-{\displaystyle \frac{n+2}{r}},1-{\displaystyle \frac{n+2}{2r}}}(Q_T)$ for $r>{\displaystyle \frac{n+2}{2}}$, one can from (4.22) gets (4.14). In addition, the estimate (3.52) implies (4.16)–(4.18).□

Proof of Theorem 1.1.

We conclude from Lemma 4.2 and Definition 2.1 that $(u,v,w)$ in Lemma 4.2 is a global weak solution of (1.1).□