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

Abstract

This paper is devoted to a p-Laplacian attraction–repulsion chemotaxis system with nonlinear sensitivity and signal production. We obtain the global existence and boundedness of weak solutions by means of energy estimates, the Aubin–Lions lemma, and parabolic and elliptic regularity estimates.

1. Introduction

Below is an introduction to a p-Laplacian chemotaxis system with nonlinear sensitivity:

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

(1)

Let $\Omega \subset {\mathbb{R}}^n$ be a smoothly bounded domain with dimension $n\ge 2$. Consider positive parameters $\chi ,\xi ,\alpha ,\beta ,\sigma ,\delta$ and an exponent $p\ge 2$. Within this framework, the unknown functions represent the following: u is cell density, v is chemoattractant concentration, and w is chemorepellant concentration. In classical linear diffusion $(p=2)$, cell motility is assumed to follow Fick’s law (where the diffusion rate is independent of density). However, in actual biological tissues, cell–cell attraction or repulsion leads to density-dependent diffusion. The p-Laplacian term $\nabla ·(|\nabla u{|}^{p-2}\nabla u)$ can describe this dependency: for $p>2$, it describes suppressed cell motility (slow diffusion), while for $1<p<2$, it describes enhanced cell motility (fast diffusion).

The smooth logistic-type source $f:\mathbb{R}\to \mathbb{R}$ satisfies for all $s>0$

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

(2)

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

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

(3)

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

Considering the effect of attraction without repulsion, problem (1) becomes the following Keller–Segel chemotaxis system:

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

(4)

In scenarios where the signal production function g exhibits linearity, specifically satisfying $g(u)=u$, it was shown that the solutions of (4) are global bounded when $n=1$, $n=2$ with ${\int }_{\Omega }{u}_{0}dx<{\displaystyle \frac{4\pi }{\chi }}$, or $n\ge 3$ with small initial conditions (see [1,2,3,4]), whereas when $n=2$ with ${\int }_{\Omega }{u}_{0}dx>{\displaystyle \frac{4\pi }{\chi }}$, or $n\ge 3$, finite-time blow-up may occur for a large class of initial data [5,6]. When the signal production $g(u)$ is nonlinear (i.e., $0<g(u)\le u^k$), the boundedness of (4) was studied in [7] if $k\in (0,{\displaystyle \frac{2}{n}})$ with $n\ge 2$. Considering the solution of (4) with nonlinear production $g(u)=u{(u+1)}^{k-1}$ under the condition $f(u)=u-\mu u^m$, it was proved in [8,9] that the weak solution of (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. [10] obtained the globally bounded classical solution of (4)= under $k<m-1$ or $k=m-1$ with $\mu >0$ sufficiently large.

Now, recall a more general system

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

(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 $D(u)$ satisfies $D(u)=D_0{u}^{\theta }$, there exists a global weak solution under the case $\theta >1-{\displaystyle \frac{2}{n}}$ [11]. Later, ref. [12] 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 shown in [13] that for the system (5) there exists a global bounded weak solution under case (i) $\xi \gamma -\chi \eta \le 0$ or case (ii) $\xi \gamma -\chi \eta >0$ with $p>{\displaystyle \frac{3n}{n+1}}$ or $1<p\le {\displaystyle \frac{3n}{n+1}}$, with $\parallel u_0{\parallel }_{L{\displaystyle \frac{(3-p)n}{p}}}$ being small. For the case $f(u)=\kappa -\mu u^m,g_1(u)=\eta u^{k_1},g_2(u)=\gamma u^{k_2}$. If ${\tau }_1=1$ and ${\tau }_2=0$, 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 (5) possesses a global bounded weak solution [14]. If ${\tau }_1={\tau }_2=1$, Jia [15] 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 [16,17,18,19,20,21].

We now formulate the principal result.

Theorem 1. 

Consider a smoothly bounded domain $\Omega \subset {\mathbb{R}}^n$ with $n\ge 2$ and an exponent $p\ge 2$. Under the assumptions that (2) and (3) hold for parameters $k_1>0$, $0<k_2\le 1$, and $m>1$, the following conclusion holds.

For arbitrary non-negative initial data $u_0,v_0,w_0\in W^{1,\infty }(\Omega )$, the problem (1) possesses a globally bounded weak solution $(u,v,w)$ in the function space $L^{\infty }(\Omega )\times W^{1,\infty }(\Omega )\times W^{1,\infty }(\Omega )$, provided either

(i)

$m>max\left\{2k_1,{\displaystyle \frac{p\alpha }{p-1}}-1\right\}$

(ii)

$k_2>max\left\{2k_1-\beta ,{\displaystyle \frac{p\alpha }{p-1}}-\beta -1\right\}$ with $m>max\{2,2\alpha ,2\beta \}$.

Remark 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 [14] 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 $m>2$.

2. Preliminaries

First, we define the weak solution of (1).

Definition 1.  

Let

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

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

When the following equation holds, $(u,v,w)$ is said to be a weak solution of (1):

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

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

and

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

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

Consider the regularity problem below:

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

(6)

for each $\epsilon \in (0,1)$. Employing a fixed-point theorem analogous to those in [18,22], the local classical solvability of system (6) can be established.

Lemma 1.  

Suppose $p\ge 2$ and $\Omega \subset {\mathbb{R}}^n$ ($n\ge 2$) is a smooth bounded domain. Given initial data $u_0,v_0,w_0\in W^{1,\infty }(\Omega )$, for each $\epsilon \in (0,1)$ there corresponds a maximal existence time $T_{max,\epsilon }\in (0,\infty ]$ and a classical solution $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$ defined locally in time. This solution lies in the space $C(\overline{\Omega }\times [0,T_{max,\epsilon }))\cap C^{2,1}(\overline{\Omega }\times [0,T_{max,\epsilon }))$ and satisfies Equation (6). When $T_{max,\epsilon }<\infty$, it holds that

$$\underset{t\to T_{max,\epsilon }}{lim}{\parallel u_{\epsilon }(·,t)\parallel }_{L^{\infty }(\Omega )}\to \infty .$$

Lemma 2.  

Assume (2) holds; $\exists M>0$ and $M:=M(u_0,|\Omega |)$ such that

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

(7)

Proof. 

Integrating the first functions of (6), we have by using (2) that

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

(8)

which implies (7). □

Lemma 3.  

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

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

(9)

satisfies

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

(10)

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

Proof. 

The proof closely resembles that of Lemma 2.2 in [23]; therefore, we will omit it. □

Next, we present the Gagliardo–Nirenberg inequality [24,25].

Lemma 4. 

Let $s>0$, $0<r\le p\le \infty$, and $q\ge 1$. Then there exists a constant $C_{GN}>0$ such that

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

(11)

where

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

(12)

3. Regularity Estimates for (6)

We study the global boundedness of the regularized problem (6).

Lemma 5.  

Given parameters $k_1>0$, $0<k_2\le 1$, and $m>1$ satisfying conditions (2) and (3), the following gradient estimate holds whenever either $m>2k_1$ or $k_2>2k_1-\beta$ with $m>max\{2,2\alpha ,2\beta \}$: there exists a positive constant $C_1$ such that for any $\epsilon \in (0,1)$,

$${\int }_{\Omega }{|\nabla v_{\epsilon }|}^{2}dx\le C_{1}forallt\in (0,T_{max}).$$

(13)

Proof. 

Case (i): $m>2k_1$.Multiplying ${(6)}_2$ by $-2▵v_{\epsilon }$ and integrating by parts, we obtain

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

(14)

Together with (14) and the first equation of (6), ∃$C_2>0$ such that

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

(15)

which used $m>2k_1$. Combining Lemma 2 and (15), we have

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

(16)

Case (ii): $k_2>2k_1-\beta$ with $m>max\{2,2\alpha ,2\beta \}$. Multiplying ${(6)}_1$ by $1+ln{u}_{\epsilon }$, we get

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

(17)

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

Multiplying ${(6)}_2$ by $-▵v_{\epsilon }$ and integrating by parts, we have

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

(18)

We estimate that

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

(19)

as well as

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

(20)

which used Lemma 3, for all $t\in (0,T_{max})$. Since $2\sigma {\int }_{\Omega }{u}_{\epsilon }ln{u}_{\epsilon }dx\le 2\sigma {\int }_{\Omega }{u}_{\epsilon }^{2}dx$, and combining (17)–(20), we have

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

(21)

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

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

(22)

Let $y(t)={\int }_{\Omega }(u_{\epsilon }ln{u}_{\epsilon }+{\displaystyle \frac{1}{2}}|\nabla v_{\epsilon }{|}^2)dx$; together with (21) and (22), this indicates

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

(23)

So we have

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

(24)

Then we derive

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

(25)

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

Lemma 6.  

Let $p\ge 2$ and assume that (2) and (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 $t\in (0,T_{max})$ and $\epsilon \in (0,1)$

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

(26)

Proof. 

Multiply ${(6)}_1$ by $q{u}_{\epsilon }^{q-1}$ and integrate over $\Omega$; thus

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

(27)

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

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

(28)

Together with (27) and (28), it follows that for $p\ge 2$

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

(29)

This is also due to

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

(30)

Quoting Lemma 3, we obtain

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

(31)

Together with (29), (30), and (31), we have

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

(32)

based on the fact that

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

(33)

From (32) and (33), we conclude that

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

(34)

Analogous to Lemma 3.11 in [26], a positive constant $C_{15}$ may be selected satisfying the condition that for all exponents $q^{\prime }\ge 2$,

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

(35)

From the combination of (34) and (35), it follows that

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

(36)

Case (i): $m>max\{2k_1,\frac{p\alpha }{p-1}-1\}$. Let $q^{\prime }:=\frac{(p-1)(q+m-1)}{(p-1)(m+1)-p\alpha }$ and take $q\ge m+3-\frac{2p\alpha }{p-1}$ and $m>\frac{p\alpha }{p-1}-1$; we have $q^{\prime }\ge 2$. Using Hölder’s inequality, we obtain

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

(37)

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

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

(38)

where $\lambda =\frac{q+m-1}{q+m-2k_1-1}>1$. We deduce from the G-N inequality that

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

(39)

for all $t\in (0,T_{max})$, where ${\theta }_1=\frac{\frac{q^{\prime }}{2}-\frac{p-1}{p}}{\frac{q^{\prime }}{2}+\frac{1}{n}-\frac{1}{2}}$. Since $p\ge 2$, selecting $q>\frac{2[(p-1)(m+1)-p\alpha ]}{p}-m+1$, we have ${\theta }_1\in (0,1)$. We invoke (39) and Lemma 5 with $m>2k_1$ such that

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

(40)

with $C_{18}>0$. We estimate again

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

(41)

with $C_{19}>0$, where ${\theta }_2=\frac{\frac{q^{\prime }}{2}-\frac{q^{\prime }}{2\lambda (q^{\prime }-1)}}{\frac{q^{\prime }}{2}+\frac{1}{n}-\frac{1}{2}}$. Due to $m>max\{1,2k_1\},p\ge 2$ and taking $q>max\{m+3-\frac{2p\alpha }{p-1},\frac{(n-2)p\alpha }{2(p-1)}-\frac{n(m-2k_1)}{2}+2-\frac{n}{2}\}$, we get ${\theta }_2\in (0,1)$. Then combining with Lemma 5, we have

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

(42)

Together with (37), (38), (40), and (42), we obtain

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

(43)

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

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

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

(44)

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

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

(45)

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

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

(46)

where

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

(47)

and

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

(48)

Obviously there is

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

(49)

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

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

(50)

This means

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

(51)

Together with (43), (45), and (51), we deduce

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

(52)

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

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

(53)

where ${\theta }_3=\frac{\frac{q^{\prime }}{2}-\frac{1}{2}}{\frac{q^{\prime }}{2}+\frac{1}{n}-\frac{1}{2}}\in (0,1)$. From (53) and Lemma 5, we conclude that

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

(54)

Due to

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

(55)

Combining (36), (52), (54), and (55), we have

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

(56)

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

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

(57)

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

Lemma 7.  

Under the assumption of Lemma 6, there exists $C_{27}>0$ such that

$$\begin{array}{c}\parallel v_{\epsilon }{(·,t)\parallel }_{W^{1,\infty }(\Omega )}\le C_{27},{\parallel w_{\epsilon }(·,t)\parallel }_{W^{1,\infty }(\Omega )}\le C_{27}forany\epsilon \in (0,1),\hfill \end{array}$$

(58)

and

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

(59)

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

Proof. 

In analogy with Lemma 3.3 of [14], we skip the proof. □

4. Proof of Theorem 1

The existence of global weak solutions to system (1) can be shown by taking limits of $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$. We start this analysis with a fundamental lemma from [13,14].

Lemma 8.  

Under the assumption of Lemma 6, there exist constants $C_1,C_2>0$ such that for all $t>0$

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

(60)

and for all $T>0$

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

(61)

Proof. 

Lemma 6’s proof establishes that ${\int }_0^t{\int }_{\Omega }{u}_{\epsilon }^{q-2}{|\nabla u_{\epsilon }|}^{p}dxdt\le C_3$. Let $q=2$; we deduce (60). In analogy with Lemma 4.1 of [14], we deduce (61). □

Lemma 9.  

Let $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$ be the classical solution of (6). There exists a non-negative funtion $(u,v,w)$ such that as $\epsilon \to 0$,

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

(62)

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

(63)

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

(64)

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

(65)

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

(66)

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

(67)

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

(68)

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

(69)

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

(70)

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

(71)

for any $T>0$.

Proof. 

Through Lemmas 7 and 8 and together with the Aubin–Lions lemma, we obtain a non-negative function u that satisfies, as $\epsilon \to 0$,

$$u_{\epsilon }\to u\mathrm{in}{L}_{\mathrm{loc}}^p(Q_T)\mathrm{and}\mathrm{almost}\mathrm{everywhere}\mathrm{in}{Q}_T.$$

(72)

Combining Lemma 7 and (72), we deduce (60). Together with (60) and (72), we obtain (65).

Collecting (64), (72) and Lemma 7, we obtain (66) by the method of Lemma 4.5 in [19]. We may further conclude from Lemma 7 in conjunction with (3) that for any $r>1$,

$$\begin{array}{c}\underset{t\in (1,\infty )}{sup}{\int }_{t-1}^t{\parallel g_1(u_{\epsilon })\parallel }_{L^r}^{r}ds\le C_4.\hfill \end{array}$$

(73)

Combining this estimate with Lemma 2.4 in [27] yields ${sup}_{t\in (1,\infty )}{\int }_{t-1}^t(\parallel v_{\epsilon }{\parallel }_{W^{2,r}}^r+\parallel v_{\epsilon t}{\parallel }_{L^r}^r)$$ds\le C_5$, which implies

$$\begin{array}{c}\parallel v_{\epsilon }{\parallel }_{W_r^{2,1}(\Omega \times (t-1,t))}\le C_6.\hfill \end{array}$$

(74)

with $C_6>0$. In view of (74), we obtain (68). Since $W_r^{2,1}(Q_T)\hookrightarrow C^{2-\frac{n+2}{r},1-\frac{n+2}{2r}}(Q_T)$ for $r>\frac{n+2}{2}$, we obtain (67). In addition, the estimate (3.46) implies (69)–(71). □

Proof of Theorem 1. 

Combining Lemma 7 and Lemma 1, we know that the system (6) has a globally bounded classical solution $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$. Then by the convergence proof of Lemma 9 and the definition of the weak solution in Definition 1, we know that the system (1) has a globally bounded weak solution $(u,v,w)$, which is the limit of $(u_{\epsilon },v_{\epsilon },w_{\epsilon })$ as $\epsilon \to 0$. □

Remark 2.  

The existence of global weak solutions rigorously proves that, under the given parameters, cell density remains finite at all times. This aligns with all empirically observed biological systems and provides a strict guarantee of solvability for complex biological chemotactic processes. Model $(1.1)$ is a new model that has never been considered in the aforementioned papers. The conclusion obtained in this paper will provide some theoretical support for the theory of partial differential equations, especially for non-Newtonian fluids.