# A new result for boundedness of solutions to a quasilinear higher-
dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Jiashan Zheng
School of Mathematics and Statistics Science,
Ludong University, Yantai 264025, P.R.China
Corresponding author. E-mail address<EMAIL_ADDRESS>(J.Zheng)

###### Abstract

This paper deals with a boundary-value problem for a coupled quasilinear
chemotaxis–haptotaxis model with nonlinear diffusion

$\left\\{\begin{array}[]{ll}u_{t}=\nabla\cdot(D(u)\nabla
u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w),\\\
\displaystyle{v_{t}=\Delta v-v+u},\\\ \displaystyle{w_{t}=-vw},\\\
\end{array}\right.$

in $N$-dimensional smoothly bounded domains, where the parameters
$\xi,\chi>0$, $\mu>0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq
C_{D}u^{m-1}$ for all $u>0$ with some $C_{D}>0$. Relying on a new energy
inequality, in this paper, it is proved that under the conditions

$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}+1)(N+\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}-1)}{N}}}}},$

and proper regularity hypotheses on the initial data, the corresponding
initial-boundary problem possesses at least one global bounded classical
solution when $D(0)>0$ (the case of non-degenerate diffusion), while if,
$D(0)\geq 0$ (the case of possibly degenerate diffusion), the existence of
bounded weak solutions for system is shown. This extends some recent results
by several authors.

Key words: Boundedness; Chemotaxis–haptotaxis; Global existence; Logistic
source

2010 Mathematics Subject Classification: 92C17, 35K55, 35K59, 35K20

##  1 Introduction

Cancer invasion is a very complex process which involves various biological
mechanisms (see [2, 18, 6, 7, 10, 15]). Chemotaxis is the oriented movement of
cells along concentration gradients of chemicals produced by the cells
themselves or in their environment, and is a significant mechanism of
directional migration of cells. A well-known chemotaxis model was proposed by
Keller and Segel ([24, 25]) in the 1970s, which describes the aggregation
processes of the cellular slime mold Dictyostelium discoideum. Since then, the
following quasi-chemotaxis-only model

$\left\\{\begin{array}[]{ll}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla v)+\mu
u(1-u),\quad x\in\Omega,t>0,\\\ \displaystyle{v_{t}=\Delta v+u-v},\quad
x\in\Omega,t>0\\\ \end{array}\right.$ (1.1)

and its variations have been widely studied by many authors, where the main
issue of the investigation was whether the solutions to the models are bounded
or blow-up (see e.g., Herrero and Velázquez [13], Nagai et al. [33], Winkler
et al. [55, 57], the survey [3]). For example, as we all known that all
solutions of (1.1) are global in time and bounded when either $N\geq 3$ and
$\mu>0$ is sufficiently large (see [54] and also [63]), or $N=2$ and $\mu>0$
is arbitrary ([34]). Tello and Winkler ([50]) proved that the global
boundedness for parabolic-elliptic chemotaxis-only system (1.1) (the second
equation of (1.1) is replaced by $-\Delta v+v=u$) exists under the condition
$\mu>\frac{(N-2)^{+}}{N}\chi$, moreover, they gave the weak solutions for
arbitrary small $\mu>0$. Some recent studies show that nonlinear chemotactic
sensitivity functions ([4, 19, 16]), nonlinear diffusion ([26, 9, 38]), or
also logistic dampening ([34, 50, 53, 54]) may prevent blow-up of solutions.

One important extension of the classical Keller-Segel model to a more complex
cell migration mechanism was proposed by Chaplain and Lolas ([7, 8]) in order
to describe processes of cancer invasion. In fact, let $u=u(x,t)$ denote the
density of the tumour cell population, $v=v(x,t)$ represent the concentration
of a matrix-degrading enzyme (MDE) and $w=w(x,t)$ stand for the density of the
surrounding tissue (extracellular matrix (ECM)). Then Chaplain and Lolas ([8])
introduced the following chemotaxis-haptotaxis system as a model describing
the process of cancer invasion

$\left\\{\begin{array}[]{ll}u_{t}=D\Delta u-\chi\nabla\cdot(u\nabla
v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w),\quad x\in\Omega,t>0,\\\
\displaystyle{\tau v_{t}=\Delta v+u-v},\quad x\in\Omega,t>0,\\\
\displaystyle{w_{t}=-vw+\eta w(1-u-w)},\quad x\in\Omega,t>0,\\\
\end{array}\right.$ (1.2)

where $\tau\in\\{0,1\\},$
$\Delta=\displaystyle{\sum_{i=1}^{N}\frac{\partial^{2}}{\partial x^{2}_{i}}}$,
$\displaystyle\frac{\partial}{\partial\nu}$ denotes the outward normal
derivative on $\partial\Omega$, $\chi>0$ and $\xi>0$ measure the chemotactic
and haptotactic sensitivities, respectively. Here $D>0$ as well as $\mu>0$ and
$\eta\geq 0$ represent the random motility coefficient, the proliferation rate
of the cells and the remodeling rate, respectively. Model (1.2) and its
analogue have been extensively studied up to now (see [43, 46, 40, 48, 47, 42,
44, 27, 5, 3, 39, 62]. In fact, Global existence and asymptotic behavior of
solutions to the haptotaxis-only system ($\chi=0$ in the first equation of
(1.2)) have been investigated in [37, 51, 30, 32] and [40] for the case
$\eta=0$ and $\eta\neq 0$, respectively.

In realistic situations, the renewal of the ECM occurs at much smaller
timescales than its degradation (see [43, 35, 22, 30, 32, 48, 51]). Therefore,
a choice of $\eta=0$ on (1.2) seems justified (see [43, 35, 22, 30, 32, 48,
51]). The models mentioned above described the random part of the motion of
cancer cells by linear diffusion, however, from a physical point of view
migration of the cancer cells through the ECM should rather be regarded like
movement in a porous medium, and so we are led to considering the cell
motility $D$ a nonlinear function of the cancer cell density. Inspired by the
analysis, in this paper,we consider the following chemotaxis-haptotaxis system
with nonlinear diffusion (see also [44, 3, 52])

$\left\\{\begin{array}[]{ll}u_{t}=\nabla\cdot(D(u)\nabla
u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w),\quad
x\in\Omega,t>0,\\\ \displaystyle{v_{t}=\Delta v+u-v},\quad x\in\Omega,t>0,\\\
\displaystyle{w_{t}=-vw},\quad x\in\Omega,t>0,\\\ \displaystyle{\frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial
w}{\partial\nu}=0},\quad x\in\partial\Omega,t>0,\\\
\displaystyle{u(x,0)=u_{0}(x)},v(x,0)=v_{0}(x),w(x,0)=w_{0}(x),\quad
x\in\Omega\\\ \end{array}\right.$ (1.3)

in a bounded domain $\Omega\subset R^{N}(N\geq 1)$ with smooth boundary
$\partial\Omega$. The origin of the system was proposed by Chaplain and Lolas
([7, 8]) to describe cancer cell invasion into surrounding healthy tissue.
Here we assume that $D(u)$ is a nonlinear function and satisfies

$D\in C^{2}([0,\infty))~{}~{}\mbox{and}~{}~{}D(u)\geq
C_{D}u^{m-1}~{}~{}\mbox{for all}~{}u>0$ (1.4)

with some $C_{D}>0$ and $m>0$. Moreover, if, $D(u)$ fulfills

$D(u)>0~{}~{}\mbox{for all}~{}u\geq 0,$ (1.5)

so the diffusion is nondegenerate and the solutions may be considered in the
sense of classical. Throughout this paper, the initial data
$(u_{0},v_{0},w_{0})$ are assumed that for some $\vartheta\in(0,1)$

$\left\\{\begin{array}[]{ll}\displaystyle{u_{0}\in
C(\bar{\Omega})~{}~{}\mbox{with}~{}~{}u_{0}\geq
0~{}~{}\mbox{in}~{}~{}\Omega~{}~{}\mbox{and}~{}~{}u_{0}\not\equiv 0},\\\
\displaystyle{v_{0}\in W^{1,\infty}(\Omega)~{}~{}\mbox{with}~{}~{}v_{0}\geq
0~{}~{}\mbox{in}~{}~{}\Omega},\\\ \displaystyle{w_{0}\in
C^{2+\vartheta}(\bar{\Omega})~{}~{}\mbox{with}~{}~{}w_{0}>0~{}~{}\mbox{in}~{}~{}\bar{\Omega}~{}~{}\mbox{and}~{}~{}\frac{\partial
w_{0}}{\partial\nu}=0~{}~{}\mbox{on}~{}~{}\partial\Omega.{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\\\
\end{array}\right.$ (1.6)

System (1.3) has been widely studied by many authors, where the main issue of
the investigation was whether the solutions to the models are bounded or blow-
up (see Tao-Winkler [44], ). For instance, when $D$ satisfies (1.4)–(1.5), Tao
and Winkler ([44]) showed that model (1.3) has global solutions provided that
$m>\max\\{1,\bar{m}\\}$, where

$\bar{m}:=\left\\{\begin{array}[]{ll}\frac{2N^{2}+4N-4}{N(N+4)}~{}~{}\mbox{if}~{}~{}N\leq
8,\\\ \frac{2N^{2}+3N+2-\sqrt{8N(N+1)}}{N(N+20)}~{}~{}\mbox{if}~{}~{}N\geq
9.\\\ \end{array}\right.$ (1.7)

However, they leave a question here: “whether the global solutions are
bounded”. If $N\geq 2$, the global boundedness of solutions to (1.3) has been
constructed for $m>2-\frac{2}{N}$ (see [27, 52]) with the help of the
boundness of $\|\nabla v\|_{L^{l}(\Omega\times(0,T))}(1\leq
l<\frac{N}{N-1})$). Recently, we ([59]) extended these results to the cases
$m>\frac{2N}{N+2}$ by using the boundness of $\|\nabla
v\|_{L^{2}(\Omega\times(0,T))}$. More recently, if $\frac{\mu}{\chi}$ is large
enough, Jin [23] (see also [20]) proved that system (1.3) admits a global
bounded solution for any $m>0$. However, we should point that the cases
$0<m\leq\frac{2N}{N+2}$ and small $\frac{\mu}{\chi}$ remain unknown even in
the case for the chemotaxis-only system (1.3), that is, $w\equiv 0$ in system
(1.3). In this paper, we firstly use the boundedness of
$\int_{(t-1)_{+}}^{t}\int_{\Omega}u^{{\gamma_{0}+1}}$ (see Lemma 3.6) for some
$\gamma_{0}>1,$ which is a new result even for chemotaxis-only system (1.3).
Then, applying the standard testing procedures, we can derive the uniform
boundedness of $\nabla v$ in $L^{l_{0}}(\Omega)$ for some $l_{0}>2.$ We
emphasize that the spontaneous boundedness information on $\nabla v$ in
$L^{l_{0}}(\Omega)$ (see (3.40)) plays a key role in this process. Using the
$L^{l_{0}}$-boundedness of $\nabla v$ and $L^{1}$-boundedness of $u$, we can
then acquire the uniform bounds of $u$ in arbitrary large $L^{p}(\Omega)$
provided that the further restriction on $m$ is satisfied (see the proof of
Lemmas 3.10-3.14). Finally, combining with Moser iteration method and
$L^{p}$-$L^{q}$ estimates for Neumann heat semigroup, we finally established
the $L^{\infty}$ bound of $u$ (see Lemmas 3.15–3.16).

Motivated by the above works, this paper will focus on studying the
relationship between the exponent $m$ and the global existence of solutions to
chemotaxis-haptotaxis model (1.3) with nonlinear diffusion. In fact, the aim
of the present paper is to study the quasilinear chemotaxis system (1.3) under
the conditions (1.4)–(1.5). For non-degenerate and degenerate diffusion both,
we will show the existence of global-in-time solutions to system (1.3) that
are uniformly bounded. The main results are as follows.

######  Theorem 1.1.

Let $\Omega\subset R^{N}(N\geq 1)$ be a bounded domain with smooth boundary
and $\chi>0,\xi>0,\mu>0$. Assume that the nonnegative initial data
$(u_{0},v_{0},w_{0})$ fulfill (1.6). Moreover, if $D$ satisfies (1.4)-(1.5)
with

$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}+1)(N+\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}-1)}{N}}}}},$
(1.8)

then there exists a triple $(u,v,w)\in(C^{0}(\bar{\Omega}\times[0,\infty))\cap
C^{2,1}(\bar{\Omega}\times(0,\infty)))^{3}$ which solves (1.3) in the
classical sense. Moreover, both $u$, $v$ and $w$ are bounded in
$\Omega\times(0,\infty)$, that is, there exists a positive constant $C$ such
that

$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|w(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
C~{}~{}~{}\mbox{for all}~{}~{}t>0.$ (1.9)

######  Remark 1.1.

(i) Obviously,

$\mu_{*}=\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{[\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu]_{+}}>1~{}~{}(\mbox{by
using}~{}~{}~{}\mu>0),$

then
$\gamma_{**}={{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{N}}}}=\mu_{*}+1+\frac{\mu_{*}^{2}-1}{N}>\mu_{*}+1>2$,
hence

$\frac{2N}{N+{{{\frac{(\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}+1)(N+\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}-1)}{N}}}}}<\frac{2N}{N+2}\leq
2-\frac{2}{N},$

therefore, Theorem 1.1 extends the results of Theorem 1.1 of Zheng ([59]), the
results of Theorem 1.1 of Wang ([52]), the results of as well as of Li-Lankeit
([27]) and partly extends the results of Theorem 1.1 of Liu et al ([31]). Here
the assumption $m>\frac{2N}{N+2}$ (see [59]) or $m>2-\frac{2}{N}$ (see [27,
31, 52]) are intrinsically required.

(ii) Obviously, for any $N\geq 1$,

$\frac{2N}{N+{{{\frac{(\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}+1)(N+\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}-1)}{N}}}}}<\bar{m},$

therefore, Theorem 1.1 extends the results of Corollary 1.2 of Tao and Winkler
([44]), who showed the global existence of solutions the cases $m>\bar{m},$
where $\bar{m}$ is given by (1.7).

(iii) In the case $N=2$, by using $\mu>0,$ then
$\frac{8}{4+(\mu_{*}+1)^{2}}<1$, our result improves the result of [41] and
[64], in which the assumption $m=1$ or $m>1$ are intrinsically required.

(iv) If $\mu>\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})$,
then by (1.8), we derive that for any $m>0,$ system (1.3) has a classical and
bounded solution, which improves the result of [23] as well as [20] and [5].

(v) The chemotaxis-haptotaxis system therefore has bounded solutions under the
same condition on $m$ as the pure chemotaxis system with $w\equiv 0$ without
logistic source (see [45]). For $\mu=0$ this condition is essentially optimal
([53]).

In the case of possibly degenerate diffusion, system (1.3) admits at least one
global bounded weak solution:

######  Theorem 1.2.

Let $\Omega\subset R^{N}(N\geq 1)$ be a bounded domain with smooth boundary
and $\chi>0,\xi>0,\mu>0$. Suppose that the initial data $(u_{0},v_{0},w_{0})$
satisfy (1.6). Moreover, if $D$ satisfies (1.4) with

$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}+1)(N+\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}-1)}{N}}}}},$

then system (1.3) admits at least one global weak solution $(u,v,w)$ in the
sense of definition 4.1 below that exists globally in time and is bounded in
the sense that (1.9) holds.

The rest of this paper is organized as follows. In the following section, we
recall some preliminary results. Section 3 is devoted to a series of a priori
estimates and then prove Theorem 1.1. In Section 4, applying the existence of
classical solutions in the non-degenerate case, we will then complete the
proof of theorem 1.2 by an approximation procedure in Section 3.

##  2 Preliminaries and main results

Before proving our main results, we will give some preliminary lemmas, which
play a crucial role in the following proofs. As for the proofs of these
lemmas, here we will not repeat them again.

######  Lemma 2.1.

([11, 21]) Let $s\geq 1$ and $q\geq 1$. Assume that $p>0$ and $a\in(0,1)$
satisfy

$\frac{1}{2}-\frac{p}{N}=(1-a)\frac{q}{s}+a(\frac{1}{2}-\frac{1}{N})~{}~{}\mbox{and}~{}~{}p\leq
a.$

Then there exist $c_{0},c^{\prime}_{0}>0$ such that for all $u\in
W^{1,2}(\Omega)\cap L^{\frac{s}{q}}(\Omega)$,

$\|u\|_{W^{p,2}(\Omega)}\leq c_{0}\|\nabla
u\|_{L^{2}(\Omega)}^{a}\|u\|^{1-a}_{L^{\frac{s}{q}}(\Omega)}+c^{\prime}_{0}\|u\|_{L^{\frac{s}{q}}(\Omega)}.$

######  Lemma 2.2.

([58]) Let $0<{\theta}\leq p\leq\frac{2N}{N-2}$. There exists a positive
constant $C_{GN}$ such that for all $u\in W^{1,2}(\Omega)\cap
L^{{\theta}}(\Omega)$,

$\|u\|_{L^{p}(\Omega)}\leq C_{GN}(\|\nabla
u\|_{L^{2}(\Omega)}^{a}\|u\|^{1-a}_{L^{{\theta}}(\Omega)}+\|u\|_{L^{{\theta}}(\Omega)})$

is valid with
$a=\displaystyle{\frac{\frac{N}{{{\theta}}}-\frac{N}{p}}{1-\frac{N}{2}+\frac{N}{{\theta}}}}\in(0,1)$.

######  Lemma 2.3.

([14, 61]) Suppose that $\gamma\in(1,+\infty)$ and $g\in
L^{\gamma}((0,T);L^{\gamma}(\Omega))$. Consider the following evolution
equation

$\left\\{\begin{array}[]{ll}v_{t}-\Delta
v+v=g,~{}~{}~{}(x,t)\in\Omega\times(0,T),\\\ \displaystyle\frac{\partial
v}{\partial\nu}=0,~{}~{}~{}(x,t)\in\partial\Omega\times(0,T),\\\
v(x,0)=v_{0}(x),~{}~{}~{}(x,t)\in\Omega.\\\ \end{array}\right.$

For each $v_{0}\in W^{2,\gamma}(\Omega)$ such that
$\displaystyle\frac{\partial v_{0}}{\partial\nu}=0$, there exists a unique
solution $v\in W^{1,\gamma}((0,T);L^{\gamma}(\Omega))\cap
L^{\gamma}((0,T);W^{2,\gamma}(\Omega)).$ In addition, if $s_{0}\in[0,T)$,
$v(\cdot,s_{0})\in W^{2,\gamma}(\Omega)(\gamma>N)$ with
$\displaystyle\frac{\partial v(\cdot,s_{0})}{\partial\nu}=0,$ then there
exists a positive constant $\lambda_{0}:=\lambda_{0}(\Omega,\gamma,N)$ such
that

$\begin{array}[]{rl}&\displaystyle{\int_{s_{0}}^{T}e^{\gamma
s}\|v(\cdot,t)\|^{\gamma}_{W^{2,\gamma}(\Omega)}ds\leq\lambda_{0}\left(\int_{s_{0}}^{T}e^{\gamma
s}\|g(\cdot,s)\|^{\gamma}_{L^{\gamma}(\Omega)}ds+e^{\gamma
s_{0}}(\|v_{0}(\cdot,s_{0})\|^{\gamma}_{W^{2,\gamma}(\Omega)})\right).}\\\
\end{array}$

The following local existence result is rather standard; since a similar
reasoning in [44, 58], see for example. Therefore, we only give the following
lemma without proof.

######  Lemma 2.4.

Assume that the nonnegative functions $u_{0},v_{0},$ and $w_{0}$ satisfies
(1.6) for some $\vartheta\in(0,1),$ $D$ satisfies (1.4) and (1.5). Then there
exists a maximal existence time $T_{max}\in(0,\infty]$ and a triple of
nonnegative functions

$(u,v,w)\in C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max}))\times
C^{0}((0,T_{max});C^{2}(\bar{\Omega}))\times
C^{2,1}(\bar{\Omega}\times[0,T_{max}))$

which solves (1.3) classically and satisfies $0\leq
w\leq\|w_{0}\|_{L^{\infty}(\Omega)}$ in $\Omega\times(0,T_{max})$. Moreover,
if $T_{max}<+\infty$, then

$\left(|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|w(\cdot,t)\|_{W^{1,\infty}(\Omega)}\right)\rightarrow\infty~{}~{}\mbox{as}~{}~{}t\nearrow
T_{max}.$ (2.1)

According to the above existence theory, for any $s\in(0,T_{max})$,
$(u(\cdot,s),v(\cdot,s),w(\cdot,s))\in C^{2}(\bar{\Omega})$. Without loss of
generality, we can assume that there exists a positive constant $K$ such that

$\|u_{0}\|_{C^{2}(\bar{\Omega})}\leq K~{}\mbox{as well as
}~{}~{}~{}\|v_{0}\|_{C^{2}(\bar{\Omega})}\leq
K~{}~{}\mbox{and}~{}~{}\|w_{0}\|_{C^{2}(\bar{\Omega})}\leq K.$ (2.2)

##  3 A priori estimates

The main task of this section is to establish for estimates for the solutions
$(u,v,w)$ of problem (1.3). To this end, in straightforward fashion one can
check the following boundedness for $u$, which is common in chemotaxis (or
chemotaxis–haptotaxis) with logistic source (see e.g. [59, 54, 52, 27]).

######  Lemma 3.1.

There exists $C>0$ such that the solution of (1.3) satisfies

$\int_{\Omega}{u}+\int_{\Omega}|\nabla v|^{2}+\int_{\Omega}|\nabla v|^{l}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})$ (3.1)

with $l\in[1,\frac{N}{N-1}).$

Since, the third component of (1.3) can be expressed explicitly in terms of
$v$. This leads to the following a one-sided pointwise estimate for $-\Delta
w$ (see e.g. [49, 41, 44]):

######  Lemma 3.2.

Let $(u,v,w)$ solve (1.3) in $\Omega\times(0,T_{max})$. Then

$\begin{array}[]{rl}-\Delta
w(x,t)\leq&\displaystyle{\|w_{0}\|_{L^{\infty}(\Omega)}\cdot
v(x,t)+\kappa~{}~{}~{}\mbox{for
all}~{}~{}x\in\Omega~{}~{}\mbox{and}~{}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$
(3.2)

where

$\kappa:=\|\Delta
w_{0}\|_{L^{\infty}(\Omega)}+4\|\nabla\sqrt{w_{0}}\|_{L^{\infty}(\Omega)}^{2}+\frac{\|w_{0}\|_{L^{\infty}(\Omega)}}{e}.$
(3.3)

Now we proceed to establish the main step towards our boundedness proof. To
this end, let us collect some basic estimates for $u$ and $v$ in comparatively
large function spaces. In fact, relying on a standard testing procedure, we
derive the following Lemma:

######  Lemma 3.3.

For any $k>1$, the solution $(u,v,w)$ of (1.3) satisfies that

$\begin{array}[]{rl}&\displaystyle{-\xi\int_{\Omega}u^{k-1}\nabla\cdot(u\nabla
w)\leq\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u^{k}vdx+\kappa\frac{({k-1})}{k}\xi\int_{\Omega}u^{k}dx,}\\\
\end{array}$ (3.4)

where $\kappa$ is the same as (3.3).

###### Proof.

For any $k>1$, we integrate the left hand of (3.4) and use Lemma 3.2 then get

$\begin{array}[]{rl}&\displaystyle{-\xi\int_{\Omega}u^{k-1}\nabla\cdot(u\nabla
w)dx}\\\ =&\displaystyle{({k-1})\xi\int_{\Omega}u^{k-1}\nabla u\cdot\nabla
wdx}\\\ =&\displaystyle{-\frac{({k-1})}{k}\xi\int_{\Omega}u^{k}\Delta wdx}\\\
\leq&\displaystyle{\frac{({k-1})}{k}\xi\int_{\Omega}u^{k}(\|w_{0}\|_{L^{\infty}(\Omega)}v+\kappa)dx,}\\\
\end{array}$ (3.5)

where $\kappa$ is the same as (3.3). This directly entails (3.4). ∎

Due to the presence of logistic source, some useful estimates for $u$ can be
derived.

######  Lemma 3.4.

(see [52, 27, 59]) Assume that $\mu>0.$ There exists a positive constant
$K_{0}$ such that the solution $(u,v,w)$ of (1.3) satisfies

$\begin{array}[]{rl}\displaystyle\int_{\Omega}u(x,t)dx\leq
K_{0}~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})\end{array}$ (3.6)

and

$\int_{t}^{t+\tau}\int_{\Omega}{u^{2}}\leq K_{0}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}-\tau),$ (3.7)

where we have set

$\tau:=\min\left\\{1,\frac{1}{6}T_{max}\right\\}.$ (3.8)

In order to establish some estimates for solution $(u,v,w)$, we first recall
the following lemma proved in [52] (see also [27, 59]).

######  Lemma 3.5.

Let $\Omega\subset\mathbb{R}^{N}(N\geq 1)$ be a bounded domain with smooth
boundary. Then for all $k>1,$ the solution $(u,v,w)$ of (1.3) satisfies that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{2}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}+\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u^{k}v+(\mu+\kappa\xi)\int_{\Omega}u^{k}}\\\
\end{array}$ (3.9)

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

###### Proof.

Multiplying $(\ref{7101.2x19x3sss189})_{1}$ (the first equation of (1.3)) by
$u^{k-1}$ and integrating over $\Omega$, we get

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+C_{D}(k-1)\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}}{}\\\ \leq&\displaystyle{-\chi\int_{\Omega}\nabla\cdot(u\nabla
v)u^{k-1}{}-\xi\int_{\Omega}\nabla\cdot(u\nabla w)u^{k-1}}{}\\\
&+\displaystyle{\mu\int_{\Omega}u^{k}(1-u-w)}{}\\\
\leq&\displaystyle{-\chi\int_{\Omega}\nabla\cdot(u\nabla
v)u^{k-1}{}-\xi\int_{\Omega}\nabla\cdot(u\nabla w)u^{k-1}}{}\\\
&+\displaystyle{\mu\int_{\Omega}u^{k}(1-u){}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.10)

according to the nonnegativity of $w$. Integrating by parts to the first term
on the right hand side of (3.10) and using the Young inequality, we obtain

$\begin{array}[]{rl}&\displaystyle{-\chi\int_{\Omega}\nabla\cdot(u\nabla
v)u^{k-1}}{}\\\ =&\displaystyle{(k-1)\chi\int_{\Omega}u^{k-1}\nabla
u\cdot\nabla v}{}\\\
\leq&\displaystyle{\frac{(k-1)C_{D}}{2}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\frac{\chi^{2}(k-1)}{2C_{D}}\int_{\Omega}u^{k+1-m}|\nabla v|^{2}.}\\\
\end{array}$ (3.11)

On the other hand, due to Lemma 3.3, we have

$\begin{array}[]{rl}&\displaystyle{-\xi\int_{\Omega}\nabla\cdot(u\nabla
w)u^{k-1}}{}\\\
\leq&\displaystyle{\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u^{k}vdx+\kappa\frac{({k-1})}{k}\xi\int_{\Omega}u^{k}dx}{}\\\
\leq&\displaystyle{\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u^{k}vdx+\kappa\xi\int_{\Omega}u^{k}dx{}.}\\\
\end{array}$ (3.12)

Furthermore, inserting (3.11)–(3.12) into (3.10), we conclude that for all
$t\in(0,T_{max})$,

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{2}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}+\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u^{k}v+(\mu+\kappa\xi)\int_{\Omega}u^{k}.}\\\
\end{array}$ (3.13)

∎

We proceed to estimate both integrals on the right of (3.9) in a
straightforward manner.

######  Lemma 3.6.

Let $(u,v,w)$ be a solution to (1.3) on $(0,T_{max})$ and

$\mu_{*}=\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{\left[\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu\right]_{+}}.$
(3.14)

If $\mu>0,$ then for all $1<\gamma_{0}<\mu_{*}$, there exists a positive
constant $C$ which depends on $\gamma_{0}$ such that

$\int_{\Omega}u^{\gamma_{0}}(x,t)\leq C~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})$ (3.15)

and

$\int_{0}^{t}\int_{\Omega}u^{\gamma_{0}+1}(x,t)\leq C~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (3.16)

###### Proof.

Multiplying $(\ref{7101.2x19x3sss189})_{1}$ by $u^{k-1}$, integrating over
$\Omega$ and using $w\geq 0$, we get

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+C_{D}(k-1)\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}dx}\\\ \leq&\displaystyle{-\chi\int_{\Omega}\nabla\cdot(u\nabla
v)u^{k-1}dx-\xi\int_{\Omega}\nabla\cdot(u\nabla
w)u^{k-1}+\mu\int_{\Omega}u^{k}(1-u-w)}\\\
\leq&\displaystyle{-\chi\int_{\Omega}\nabla\cdot(u\nabla
v)u^{k-1}dx-\xi\int_{\Omega}\nabla\cdot(u\nabla
w)u^{k-1}+\mu\int_{\Omega}u^{k}(1-u).}\\\ \end{array}$ (3.17)

We now estimate the right hand side of (3.17) terms by terms. To this end,
integrating by parts to the first term on the right hand side of (3.17), we
obtain for any $\varepsilon_{1}>0,$

$\begin{array}[]{rl}&\displaystyle{-\chi\int_{\Omega}\nabla\cdot(u\nabla
v)u^{k-1}}\\\ =&\displaystyle{-\frac{({k-1})\chi}{k}\int_{\Omega}u^{k}\Delta
v}\\\ \leq&\displaystyle{\frac{(k-1)\chi}{k}\int_{\Omega}u^{k}|\Delta v|}\\\
\leq&\displaystyle{\varepsilon_{1}\int_{\Omega}u^{k+1}+\gamma_{1}\varepsilon_{1}^{-k}\int_{\Omega}|\Delta
v|^{k+1},}\\\ \end{array}$ (3.18)

where

$\gamma_{1}=\frac{1}{k+1}\left(\frac{k+1}{k}\right)^{-k}\left(\frac{(k-1)\chi}{k}\right)^{k+1}.$
(3.19)

Due to (3.2) and (3.3), it follows that for any $\varepsilon_{2}>0$

$\begin{array}[]{rl}&\displaystyle{-\xi\int_{\Omega}\nabla\cdot(u\nabla
w)u^{k-1}}\\\ =&\displaystyle{-\frac{({k-1})\xi}{k}\int_{\Omega}u^{k}\Delta
w}\\\
\leq&\displaystyle{\kappa\frac{({k-1})\xi}{k}\int_{\Omega}u^{k}+\frac{({k-1})\xi\|w_{0}\|_{L^{\infty}(\Omega)}}{k}\int_{\Omega}u^{k}v}\\\
\leq&\displaystyle{\kappa\xi\int_{\Omega}u^{k}+\frac{({k-1})\xi\|w_{0}\|_{L^{\infty}(\Omega)}}{k}\int_{\Omega}u^{k}v}\\\
\leq&\displaystyle{\kappa\xi\int_{\Omega}u^{k}+\varepsilon_{2}\int_{\Omega}u^{k+1}+\gamma_{2}\varepsilon_{2}^{-k}\int_{\Omega}v^{k+1},}\\\
\end{array}$ (3.20)

where

$\gamma_{2}:=\frac{1}{k+1}\left(\frac{k+1}{k}\right)^{-k}\left(\frac{({k-1})\xi\|w_{0}\|_{L^{\infty}(\Omega)}}{k}\right)^{k+1}$
(3.21)

and $\kappa$ is give by (3.3).

On the other hand, in view of $k>1$, we also derive that

$\begin{array}[]{rl}\mu\displaystyle\int_{\Omega}u^{k}(1-u)=&\displaystyle{-\mu\int_{\Omega}u^{k+1}+(\mu+\frac{k+1}{k})\int_{\Omega}u^{k}-\frac{k+1}{k}\int_{\Omega}u^{k}}\\\
\leq&\displaystyle{-\mu\int_{\Omega}u^{k+1}+(\mu+2)\int_{\Omega}u^{k}-\frac{k+1}{k}\int_{\Omega}u^{k}.}\\\
\end{array}$ (3.22)

Therefore, combined with (3.18), (3.20), (3.17) as well as (3.22) and (1.5),
we have

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+C_{D}(k-1)\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}+\frac{k+1}{k}\int_{\Omega}u^{k}}\\\
\leq&\displaystyle{(-\mu+\varepsilon_{1}+\varepsilon_{2})\int_{\Omega}u^{k+1}+\gamma_{1}\varepsilon_{1}^{-k}\int_{\Omega}|\Delta
v|^{k+1}+\gamma_{2}\varepsilon_{2}^{-k}\int_{\Omega}v^{k+1}+C_{1}\int_{\Omega}u^{k}}\\\
\end{array}$ (3.23)

with $C_{1}=\kappa\xi+\mu+2.$ For any $t\in(0,T_{max})$, applying the Gronwall
Lemma to the above inequality shows that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{k}}\|u(\cdot,t)\|^{{{k}}}_{L^{{k}}(\Omega)}+C_{D}(k-1)\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}}\\\
\leq&\displaystyle{\frac{1}{{k}}e^{-({{k}+1})t}\|u_{0}\|^{{{k}}}_{L^{{k}}(\Omega)}+(\varepsilon_{1}+\varepsilon_{2}-\mu)\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}+1}}dxds}\\\
&+\displaystyle{\gamma_{1}\varepsilon_{1}^{-k}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}|\Delta
v|^{{k}+1}dxds+C_{1}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}}}dxds}\\\
&\displaystyle{+\gamma_{2}\varepsilon_{2}^{-k}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}v^{{{k}+1}}dxds}\\\
\leq&\displaystyle{(\varepsilon_{1}+\varepsilon_{2}-\mu)\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}+1}}dxds+\gamma_{1}\varepsilon_{1}^{-k}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}|\Delta
v|^{{k}+1}dxds}\\\
&+\displaystyle{\gamma_{2}\varepsilon_{2}^{-k}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}v^{{{k}+1}}dxds+C_{1}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}}}dxds+C_{2},}\\\
\end{array}$ (3.24)

where

$C_{2}:=C_{2}({k})=\frac{1}{{k}}\|u_{0}\|^{{{k}}}_{L^{{k}}(\Omega)}.$

Next, a use of Lemma 2.3 and (2.2) leads to

$\begin{array}[]{rl}&\displaystyle{\gamma_{1}\varepsilon_{1}^{-k}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}|\Delta
v|^{{k}+1}dxds}\\\
=&\displaystyle{\gamma_{1}\varepsilon_{1}^{-k}e^{-({{k}+1})t}\int_{0}^{t}e^{({{k}+1})s}\int_{\Omega}|\Delta
v|^{{k}+1}dxds}\\\
\leq&\displaystyle{\gamma_{1}\varepsilon_{1}^{-k}e^{-({{k}+1})t}\lambda_{0}(\int_{0}^{t}\int_{\Omega}e^{({{k}+1})s}u^{{k}+1}dxds+\|v_{0}\|^{{k}+1}_{W^{2,{{k}+1}}(\Omega)})}\\\
\end{array}$ (3.25)

and

$\begin{array}[]{rl}&\displaystyle{\gamma_{2}\varepsilon_{2}^{-k}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}v^{{k}+1}dxds}\\\
=&\displaystyle{\gamma_{2}\varepsilon_{2}^{-k}e^{-({{k}+1})t}\int_{0}^{t}e^{({{k}+1})s}\int_{\Omega}v^{{k}+1}dxds}\\\
\leq&\displaystyle{\gamma_{2}\varepsilon_{2}^{-k}e^{-({{k}+1})t}\lambda_{0}(\int_{0}^{t}\int_{\Omega}e^{({{k}+1})s}u^{{k}+1}dxds+\|v_{0}\|^{{k}+1}_{W^{2,{{k}+1}}(\Omega)})}\\\
\end{array}$ (3.26)

for all $t\in(0,T_{max})$, where $\lambda_{0}$ is the same as Lemma 2.3. On
the other hand, choosing
$\varepsilon_{1}=\frac{(k-1)\chi}{k+1}\lambda_{0}^{\frac{1}{k+1}}$ and
$\varepsilon_{2}=\frac{(k-1)\xi\|w_{0}\|_{L^{\infty}(\Omega)}}{k+1}\lambda_{0}^{\frac{1}{k+1}}$,
with the help of (3.19) and (3.21), a simple calculation shows that

$\varepsilon_{1}+\gamma_{1}\lambda_{0}\varepsilon_{1}^{-k}=\frac{({k}-1)}{{k}}\lambda_{0}^{\frac{1}{{k}+1}}\chi$

and

$\varepsilon_{2}+\gamma_{2}\lambda_{0}\varepsilon_{2}^{-k}=\frac{({k}-1)}{{k}}\lambda_{0}^{\frac{1}{{k}+1}}\xi\|w_{0}\|_{L^{\infty}(\Omega)},$

so that, substituting (3.25)–(3.26) into (3.24) implies that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{k}}\|u(\cdot,t)\|^{{{k}}}_{L^{{k}}(\Omega)}+C_{D}(k-1)\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}}\\\
\leq&\displaystyle{(\varepsilon_{1}+\gamma_{1}\lambda_{0}\varepsilon_{1}^{-k}+\varepsilon_{2}+\gamma_{2}\lambda_{0}\varepsilon_{2}^{-k}-\mu)\int_{0}^{t}e^{-({k}+1)(t-s)}\int_{\Omega}u^{{{k}+1}}dxds}\\\
&+\displaystyle{(\gamma_{1}\varepsilon_{1}^{-k}+\gamma_{2}\varepsilon_{2}^{-k})e^{-({k}+1)(t-s_{0})}\lambda_{0}\|v_{0}\|^{{k}+1}_{W^{2,{{k}+1}}(\Omega)}+C_{1}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}}}dxds+C_{2}}\\\
=&\displaystyle{(\frac{({k}-1)}{{k}}\lambda_{0}^{\frac{1}{{k}+1}}\chi+\frac{({k}-1)}{{k}}\lambda_{0}^{\frac{1}{{k}+1}}\xi\|w_{0}\|_{L^{\infty}(\Omega)}-\mu)\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}+1}}dxds}\\\
&+\displaystyle{(\gamma_{1}\varepsilon_{1}^{-k}+\gamma_{2}\varepsilon_{2}^{-k})e^{-({k}+1)(t-s_{0})}\lambda_{0}\|v_{0}\|^{{k}+1}_{W^{2,{{k}+1}}(\Omega)}+C_{1}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}}}dxds+C_{2}}\\\
\leq&\displaystyle{[\frac{({k}-1)}{{k}}\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu]\int_{0}^{t}e^{-({k}+1)(t-s)}\int_{\Omega}u^{{{k}+1}}}\\\
&+\displaystyle{C_{1}\int_{0}^{t}e^{-({{k}+1})(t-s)}\int_{\Omega}u^{{{k}}}dxds+C_{3}}\\\
\end{array}$ (3.27)

with

$C_{3}=(\gamma_{1}\varepsilon_{1}^{-k}+\gamma_{2}\varepsilon_{2}^{-k})e^{-({k}+1)(t-s_{0})}\lambda_{0}\|v_{0}\|^{{k}+1}_{W^{2,{{k}+1}}}+C_{2}.$

For any $\varepsilon>0,$ we choose $k=\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}-\varepsilon.$
Then

$\frac{({k}-1)}{{k}}\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})<\mu.$

Thus, by using the Young inequality, we derive that there exists a positive
constant $C_{4}$ such that

$\begin{array}[]{rl}&\displaystyle{\int_{\Omega}u^{{k}}(x,t)dx\leq
C_{4}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.28)

and

$\int_{0}^{t}\int_{\Omega}u^{k+1}(x,t)\leq C_{4}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (3.29)

Thereupon, combining with the arbitrariness of $\varepsilon$ and the Hölder
inequality, (3.15) and (3.16) holds. The proof of Lemma 3.6 is completed. ∎

When

$\mu\geq\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)}),$

by making use of above lemma, we can derive the following results on the bound
$u$ for in an $L^{k}$ space for any $k>1$.

######  Corollary 3.1.

Let $(u,v,w)$ be a solution to (1.3) on $(0,T_{max})$. If

$\mu\geq\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)}),$

then for all $k>1$, there exists a positive constant $C$ which depends on $k$
such that

$\int_{\Omega}u^{k}(x,t)\leq C~{}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}).$
(3.30)

###### Proof.

This directly results from Lemma 3.6 and the fact that

$\frac{\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})}{(\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})-\mu)_{+}}=+\infty$

by using $\mu\geq\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)}).$ ∎

In the following, we always assume that

$\mu<\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)}),$

since, case $\mu\geq\max_{s\geq
1}\lambda_{0}^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_{0}\|_{L^{\infty}(\Omega)})$ has
been proved by Corollary 3.1.

######  Lemma 3.7.

Let $(u,v,w)$ be a solution to (1.3) on $(0,T_{max})$ and
$\Omega\subset\mathbb{R}^{N}(N\geq 1)$ be a bounded domain with smooth
boundary. Then for all $\beta>1,$ there exists $\kappa_{0}>0$ such that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{2\beta}}\frac{d}{dt}\|\nabla
v\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)}+\frac{(\beta-1)}{{\beta^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla
v|^{\beta}\right|^{2}}\\\
&+\displaystyle{\frac{1}{2}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}+\displaystyle\int_{\Omega}|\nabla v|^{2\beta}}{}\\\
\leq&\displaystyle{\kappa_{0}\int_{\Omega}u^{2}|\nabla
v|^{2\beta-2}+\kappa_{0}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}).}\\\
\end{array}$ (3.31)

###### Proof.

Using that $\nabla v\cdot\nabla\Delta{v}=\frac{1}{2}\Delta|\nabla
v|^{2}-|D^{2}v|^{2}$, by a straightforward computation using the second
equation in (1.3) and several integrations by parts, we find that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{2\beta}}\frac{d}{dt}\|\nabla
v\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)}}\\\
=&\displaystyle{\displaystyle\int_{\Omega}|\nabla v|^{2\beta-2}\nabla
v\cdot\nabla(\Delta v-v+{u})}\\\
=&\displaystyle{\frac{1}{{2}}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}\Delta|\nabla v|^{2}-\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}}\\\ &-\displaystyle\int_{\Omega}|\nabla
v|^{2\beta}-\displaystyle{\displaystyle\int_{\Omega}u\nabla\cdot(|\nabla
v|^{2\beta-2}\nabla v)}\\\
=&\displaystyle{-\frac{\beta-1}{{2}}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-4}\left|\nabla|\nabla
v|^{2}\right|^{2}+\frac{1}{{2}}\displaystyle\int_{\partial\Omega}|\nabla
v|^{2\beta-2}\frac{\partial|\nabla
v|^{2}}{\partial\nu}-\displaystyle\int_{\Omega}|\nabla v|^{2\beta}}\\\
&-\displaystyle{\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}-\displaystyle\int_{\Omega}{u}|\nabla
v|^{2\beta-2}\Delta v-\displaystyle\int_{\Omega}u\nabla v\cdot\nabla(|\nabla
v|^{2\beta-2})}\\\
=&\displaystyle{-\frac{2(\beta-1)}{{\beta^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla
v|^{\beta}\right|^{2}+\frac{1}{{2}}\displaystyle\int_{\partial\Omega}|\nabla
v|^{2\beta-2}\frac{\partial|\nabla
v|^{2}}{\partial\nu}-\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}}\\\
&-\displaystyle{\displaystyle\int_{\Omega}{u}|\nabla v|^{2\beta-2}\Delta
v-\displaystyle\int_{\Omega}u\nabla v\cdot\nabla(|\nabla
v|^{2\beta-2})-\displaystyle\int_{\Omega}|\nabla v|^{2\beta}}\\\ \end{array}$
(3.32)

for all $t\in(0,T_{max})$. Here, since $|\Delta v|\leq\sqrt{N}|D^{2}v|$, by
the Young inequality, we can estimate

$\begin{array}[]{rl}\displaystyle\int_{\Omega}{u}|\nabla v|^{2\beta-2}\Delta
v\leq&\displaystyle{\sqrt{N}\displaystyle\int_{\Omega}{u}|\nabla
v|^{2\beta-2}|D^{2}v|}\\\
\leq&\displaystyle{\frac{1}{4}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}+N\displaystyle\int_{\Omega}u^{2}|\nabla
v|^{2\beta-2}}\\\ \end{array}$ (3.33)

for all $t\in(0,T_{max})$. As moreover by the Cauchy–Schwarz inequality, we
have

$\begin{array}[]{rl}-\displaystyle\int_{\Omega}u\nabla v\cdot\nabla(|\nabla
v|^{2\beta-2})=&\displaystyle{-(\beta-1)\displaystyle\int_{\Omega}{u}|\nabla
v|^{2(\beta-2)}\nabla v\cdot\nabla|\nabla v|^{2}}\\\
\leq&\displaystyle{\frac{\beta-1}{8}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-4}\left|\nabla|\nabla
v|^{2}\right|^{2}+2(\beta-1)\displaystyle\int_{\Omega}u^{2}|\nabla
v|^{2\beta-2}}\\\
\leq&\displaystyle{\frac{(\beta-1)}{2{\beta^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla
v|^{\beta}\right|^{2}+2(\beta-1)\displaystyle\int_{\Omega}u^{2}|\nabla
v|^{2\beta-2}.}\\\ \end{array}$ (3.34)

Next we deal with the integration on $\partial\Omega$. We see from Lemma 2.1
that

$\begin{array}[]{rl}&\displaystyle{\displaystyle\int_{\partial\Omega}\frac{\partial|\nabla
v|^{2}}{\partial\nu}|\nabla v|^{2\beta-2}}\\\
\leq&\displaystyle{C_{\Omega}\displaystyle\int_{\partial\Omega}|\nabla
v|^{2\beta}}\\\ =&\displaystyle{C_{\Omega}||\nabla
v|^{\beta}|^{2}_{L^{2}(\partial\Omega)}.}\\\ \end{array}$ (3.35)

Let us take $r\in(0,\frac{1}{2})$. By the embedding
$W^{r+\frac{1}{2},2}(\Omega)\hookrightarrow L^{2}(\partial\Omega)$ is compact
(see e.g. Haroske and Triebel [12]), we have

$\begin{array}[]{rl}&\displaystyle{\||\nabla
v|^{\beta}\|^{2}_{L^{2}{(\partial\Omega})}\leq C_{3}\||\nabla
v|^{\beta}\|^{2}_{W^{r+\frac{1}{2},2}(\Omega)}.}\\\ \end{array}$ (3.36)

In order to apply Lemma 2.1 to the right-hand side of (3.36), let us pick
$a\in(0,1)$ satisfying

$a=\frac{\frac{1}{2N}+\frac{\beta}{l}+\frac{\gamma}{N}-\frac{1}{2}}{\frac{1}{N}+\frac{\beta}{l}-\frac{1}{2}}.$

Noting that $\gamma\in(0,\frac{1}{2})$ and $\beta>1$ imply that
$\gamma+\frac{1}{2}\leq a<1$, we see from the fractional Gagliardo–Nirenberg
inequality (Lemma 2.1) and boundedness of $|\nabla v|^{l}$ (see Lemma 3.1)
that

$\begin{array}[]{rl}&\displaystyle{||\nabla
v|^{\beta}|^{2}_{W^{r+\frac{1}{2},2}(\Omega)}}\\\
\leq&\displaystyle{c_{0}|\nabla|\nabla
v|^{\beta}|^{a}_{L^{2}(\Omega)}\||\nabla
v|^{\beta}\|^{1-a}_{L^{\frac{l}{\beta}}(\Omega)}+c^{\prime}_{0}\||\nabla
v|^{\beta}\|_{L^{\frac{l}{\beta}}(\Omega)}}\\\
\leq&\displaystyle{C_{4}|\nabla|\nabla
v|^{\beta}|^{a}_{L^{2}(\Omega)}+C_{4}.}\\\ \end{array}$ (3.37)

Combining (3.35) and (3.36) with (3.37), we obtain

$\begin{array}[]{rl}&\displaystyle{\displaystyle\int_{\partial\Omega}\frac{\partial|\nabla
v|^{2}}{\partial\nu}|\nabla v|^{2\beta-2}\leq C_{5}|\nabla|\nabla
v|^{\beta}|^{a}_{L^{2}(\Omega)}+C_{5}.}\\\ \end{array}$ (3.38)

Now, inserting (3.34)–(3.38) into (3.32) and using the Young inequality we can
get

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{2\beta}}\frac{d}{dt}\|\nabla
v\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)}+\frac{3(\beta-1)}{4{\beta^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla
v|^{\beta}\right|^{2}+\frac{1}{2}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}+\displaystyle\int_{\Omega}|\nabla v|^{2\beta}}\\\
\leq&\displaystyle{C_{6}\displaystyle\int_{\Omega}u^{2}|\nabla
v|^{2\beta-2}+C_{6}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$
(3.39)

by using the Young inequality. ∎

We proceed to establish the main step towards our boundedness proof. The
following lemma can be used to improve our knowledge on integrability of
$\nabla v$, provided that $\mu>0$. Its repeated application will form the core
of our regularity proof.

######  Lemma 3.8.

Let $(u,v,w)$ be a solution to (1.3) on $(0,T_{max})$ and $\mu>0$. Then for
any $1<\gamma_{0}<\mu_{*}$, there exists $C>0$ such that

$\|\nabla
v(\cdot,t)\|_{L^{{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}}(\Omega)}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}),$ (3.40)

where $\mu_{*}$ is given by (3.14).

###### Proof.

Let $\gamma_{0}$ and $\mu_{*}$ be same as Lemma 3.6. For the above
$1<\gamma_{0}<\mu_{*}$, we choose $\beta=\frac{\gamma_{0}+1}{2}$ in (3.31).
Then by using the Young inequality, we derive that for some positive constant
$C_{1}$,

$\begin{array}[]{rl}\displaystyle{\kappa_{0}\int_{\Omega}{u^{2}}|\nabla{v}|^{2\beta-2}}=&\displaystyle{\kappa_{0}\int_{\Omega}{u^{2}}|\nabla{v}|^{\gamma_{0}-1}}\\\
\leq&\displaystyle{\frac{1}{2}\int_{\Omega}|\nabla{v}|^{\gamma_{0}+1}+C_{1}\int_{\Omega}{u}^{\gamma_{0}+1}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).}\\\ \end{array}$ (3.41)

Here $\kappa_{0}$ is the same as (3.31). Inserting (3.41) into (3.31), we
conclude that there exists a positive constant $C_{2}$ such that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{\gamma_{0}+1}}\frac{d}{dt}\|\nabla{v}\|^{{{\gamma_{0}+1}}}_{L^{{\gamma_{0}+1}}(\Omega)}+\frac{3(\frac{\gamma_{0}+1}{2}-1)}{{(\gamma_{0}+1)^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla{v}|^{\frac{\gamma_{0}+1}{2}}\right|^{2}}\\\
&\displaystyle{+\frac{1}{2}\displaystyle\int_{\Omega}|\nabla{v}|^{\gamma_{0}-1}|D^{2}{v}|^{2}+\displaystyle\frac{1}{2}\int_{\Omega}|\nabla{v}|^{\gamma_{0}+1}}\\\
\leq&\displaystyle{C_{1}\int_{\Omega}{u^{\gamma_{0}+1}}+C_{2}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (3.42)

which combined with (3.16) implies that

$\int_{\Omega}|\nabla v|^{\gamma_{0}+1}(x,t)dx\leq C_{3}~{}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})$ (3.43)

by an ODE comparison argument. On the other hand, for any $\beta>1$, it then
follows from Lemma 2.2 that there exist positive constants $\kappa_{1}$ and
$\kappa_{2}$ such that

$\begin{array}[]{rl}\|\nabla{v}\|_{L^{2\beta+\frac{2(\gamma_{0}+1)}{N}}(\Omega)}^{2\beta+\frac{2(\gamma_{0}+1)}{N}}=&\displaystyle{\||\nabla{v}|^{\beta}\|_{L^{2+\frac{2(\gamma_{0}+1)}{\beta
N}}(\Omega)}^{2+\frac{2(\gamma_{0}+1)}{N\beta}}}\\\
\leq&\displaystyle{\kappa_{1}(\|\nabla|\nabla{v}|^{\beta}\|_{L^{2}(\Omega)}^{2}\||\nabla{v}|^{\beta}\|_{L^{\frac{\gamma_{0}+1}{\beta}}(\Omega)}^{\frac{2(\gamma_{0}+1)}{N\beta}}+\||\nabla{v}|^{\beta}\|_{L^{\frac{\gamma_{0}+1}{\beta}}(\Omega)}^{2+\frac{2(\gamma_{0}+1)}{N\beta}})}\\\
\leq&\displaystyle{\kappa_{2}(\|\nabla|\nabla{v}|^{\beta}\|_{L^{2}(\Omega)}^{2}+1)}\\\
\end{array}$ (3.44)

by using (3.43). Next, picking
$\beta=\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{2N}$ in (3.31), then $\beta>1$,
so that, by (3.31), we derive that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}\frac{d}{dt}\|\nabla{v}\|^{{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}}_{L^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}(\Omega)}}\\\
&\displaystyle{+\frac{3({\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{2N}}-1)}{{({\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}})^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla{v}|^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{2N}}}\right|^{2}}\\\
&\displaystyle{+\frac{1}{2}\displaystyle\int_{\Omega}|\nabla{v}|^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}-2}|D^{2}{v}|^{2}+\displaystyle\int_{\Omega}|\nabla{v}|^{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}\\\
\leq&\displaystyle{\kappa_{0}\displaystyle\int_{\Omega}{u^{2}}|\nabla{v}|^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}-2}+\kappa_{0}}\\\
\leq&\displaystyle{\frac{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{2N}}-1}{{({\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}})^{2}\kappa_{2}}}\|\nabla{v}\|_{L^{2\beta+\frac{2(\gamma_{0}+1)}{N}}(\Omega)}^{2\beta+\frac{2(\gamma_{0}+1)}{N}}+C_{4}\displaystyle\int_{\Omega}{u^{\gamma_{0}+1}}+C_{5}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (3.45)

where $\kappa_{2}$ is the same as (3.44). Therefore, collecting (3.44) and
(3.45), we have

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}\frac{d}{dt}\|\nabla{v}\|^{{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}}_{L^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}(\Omega)}}\\\
&\displaystyle{+\frac{2({\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{2N}}-1)}{{({\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}})^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla{v}|^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{2N}}}\right|^{2}}\\\
&\displaystyle{+\frac{1}{2}\displaystyle\int_{\Omega}|\nabla{v}|^{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}-2}|D^{2}{v}|^{2}+\displaystyle\int_{\Omega}|\nabla{v}|^{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}\\\
\leq&\displaystyle{C_{4}\displaystyle\int_{\Omega}{u^{\gamma_{0}+1}}+C_{5}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (3.46)

therefore, in view of (3.16), by using an ODE comparison argument again, we
have

$\begin{array}[]{rl}\|\nabla{v}\|_{L^{{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}}(\Omega)}\leq&\displaystyle{C_{6}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.47)

with some positive constant $C_{6}$, which yields (3.40), and hence completes
the proof. ∎

######  Lemma 3.9.

Let $\Omega\subset\mathbb{R}^{N}(N\geq 1)$ be a bounded domain with smooth
boundary. Then for all $\beta>1$ and $k>1$, the solution of (1.3) from Lemma
2.4 satisfies

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}(\frac{1}{k}\|u\|^{k}_{L^{k}(\Omega)}+\frac{1}{{2\beta}}\|\nabla{v}\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)})+\frac{3(\beta-1)}{4{\beta^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla{v}|^{\beta}\right|^{2}+\frac{\mu}{2}\int_{\Omega}u^{k+1}}\\\
&+\displaystyle{\frac{1}{2}\displaystyle\int_{\Omega}|\nabla{v}|^{2\beta-2}|D^{2}{v}|^{2}+\displaystyle\int_{\Omega}|\nabla{v}|^{2\beta}+\frac{(k-1)m}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}}{}\\\
\leq&\displaystyle{C(\displaystyle\frac{\chi^{2}(k-1)}{2C_{D}}\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}+\int_{\Omega}u^{2}|\nabla{v}|^{2\beta-2}+\int_{\Omega}v^{k+1})+C,}\\\
\end{array}$ (3.48)

where $C$ is a positive constant.

###### Proof.

Collecting Lemma 3.5 and Lemma 3.8, we can derive (3.48) by using the Young
inequality. ∎

We next plan to estimate the right-hand sides in the above inequalities
appropriately by using a priori information provided by Lemma 3.8 and Lemma
3.1. Here the following lemma will will play an important role in making
efficient use of the known
${L^{{{\frac{(\gamma_{0}+1)(N+\gamma_{0}-1)}{N}}}}(\Omega)}$ bound for $\nabla
v$. The following lemma provides some elementary material that will be
essential to our bootstrap procedure.

######  Lemma 3.10.

Let

$\tilde{H}(y)=\begin{array}[]{ll}\frac{2N^{2}}{N^{2}+{{{(y+1)(N+y-1)}}}}-[1+\frac{[N^{2}-{{{(y+1)(N+y-1)}}}]y}{N{{{(y+1)(N+y-1)}}}}]\end{array}$
(3.49)

with ${{{(y+1)(N+y-1)}}}>N^{2}$, for any $y>1$ and $N\geq 2.$ Then we have

$\min_{y>1}\tilde{H}(y)\geq 0.$ (3.50)

###### Proof.

It is easy to verify that $N^{2}<{{{(y+1)(N+y-1)}}}$ and $y>1$ and $N\geq 2$
implies that

$y>\frac{-N+\sqrt{5N^{2}-4N+4}}{2}=\frac{-N+\sqrt{4N^{2}+(N-2)^{2}}}{2}\geq\frac{N}{2}.$
(3.51)

On the other hand, by some basic calculation, one has

$\begin{array}[]{ll}&\frac{2N^{2}}{N^{2}+{{{(y+1)(N+y-1)}}}}-[1+\frac{[N^{2}-{{{(y+1)(N+y-1)}}}]y}{N{{{(y+1)(N+y-1)}}}}]\\\
=&[{{{(y+1)(N+y-1)}}}-N^{2}][\frac{y}{N{{{(y+1)(N+y-1)}}}}-\frac{1}{N^{2}+{{{(y+1)(N+y-1)}}}}]\\\
=&\frac{[{{{(y+1)(N+y-1)}}}-N^{2}]}{N{{{(y+1)(N+y-1)[N^{2}+{{{(y+1)(N+y-1)}}}}}]}}h_{1}(y)\\\
\end{array}$ (3.52)

with

$\begin{array}[]{rl}h_{1}(y)=&[yN^{2}+y(y+1)(y+N-1)-N(y+1)(N+y-1)]\\\
=&y^{3}+(N-1)y-N^{2}+N.\\\ \end{array}$ (3.53)

Now, by some basic calculation, one has,

$\begin{array}[]{rl}h^{\prime}_{1}(y)=&3y^{2}+(N-1)>0.\\\ \end{array}$

by using $N\geq 2$ and $y>1$. Therefore, by (3.51), we have

$\begin{array}[]{rl}h_{1}(y)\geq&h_{1}(\frac{N}{2})\\\
=&\frac{N^{3}}{8}-\frac{N^{2}}{2}+\frac{N}{2}\\\ =:&\tilde{h}_{1}(N).\\\
\end{array}$ (3.54)

Since, $\tilde{h}^{\prime}_{1}(N)=\frac{3N^{2}}{8}-N+\frac{1}{2}>0$ by using
$N\geq 2.$ Thus, $\tilde{h}_{1}(N)>\tilde{h}_{1}(2)=0,$ so that, inserting
(3.53)–(3.54) into (3.52) and ${{{(y+1)(N+y-1)}}}>N^{2}$, we obtain that

$\frac{2N^{2}}{N^{2}+{{{(y+1)(N+y-1)}}}}>[1+\frac{[N^{2}-{{{(y+1)(N+y-1)}}}]y}{N{{{(y+1)(N+y-1)}}}}].$
(3.55)

∎

Now, we can make use of Lemma 3.4 as well as Lemma 2.2 and Lemma 3.8 to
estimate the integrals on the right-hand sides of (3.31) and (3.9) (or
(3.48)). To this end, we will establish bounds for $\int_{\Omega}u^{k}dx$ with
any $k>1$ by Lemma 3.4 as well as Lemma 2.2 and Lemma 3.10.

######  Lemma 3.11.

Assume that $m>\frac{2N}{N+\gamma_{*}}$ with $N=2$, where

$\gamma_{*}={{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{N}}}}$ (3.56)

and $\mu_{*}$ is same as (3.14). Then for all $k>1$, there exists $C>0$ such
that

$\|u(\cdot,t)\|_{L^{k}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (3.57)

###### Proof.

Next, due to (3.56) and $N=2$ and $m>\frac{2N}{N+\gamma_{*}}$ implies that

$m>\frac{8}{4+[\mu_{*}+1]^{2}},$ (3.58)

where $\mu_{*}$ is given by (3.14). Now, in view of $\mu>0$ implies that

$\mu_{*}>1$

and

$4<(\mu_{*}+1)^{2},$

therefore, employing Lemma 3.10, we have

$m>1+\frac{[1-\frac{(\mu_{*}+1)^{2}}{4}]\mu_{*}}{2\times\frac{(\mu_{*}+1)^{2}}{4}}.$
(3.59)

Thus, we may choose $q_{0}\in(1,\mu_{*})$ which is close to $\mu_{*}$ such
that

$\begin{array}[]{rl}m>&1+\frac{[1-\frac{(q_{0}+1)^{2}}{4}]q_{0}}{2\times\frac{(q_{0}+1)^{2}}{4}}.\\\
\end{array}$ (3.60)

Next, observing that
${{{{\frac{(q_{0}+1)^{2}}{4}}}}}\in(1,\frac{(\mu_{*}+1)^{2}}{4})$, thus, in
light of Lemma 3.8, we derive that there exists a positive constant $C_{1}$
such that

$\|\nabla v(\cdot,t)\|_{L^{2p_{0}}(\Omega)}\leq C_{1}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (3.61)

where

$p_{0}={{{{\frac{(q_{0}+1)^{2}}{4}}}}}>1$ (3.62)

by using $q_{0}>1.$

Now, choosing $k>\max\\{3,|1-m|+q_{0}\frac{p_{0}-1}{p_{0}}\\}$ in (3.9), then,
we have

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{2}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}+C_{2}\int_{\Omega}u^{k}(v+1)}\\\ \end{array}$ (3.63)

with $C_{2}=\max\\{\xi\|w_{0}\|_{L^{\infty}(\Omega)},\mu+\kappa\xi\\}$. We
estimate the rightmost integral by means of the Hölder inequality according to

$\begin{array}[]{rl}&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\displaystyle\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}}\\\
&\leq\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\left(\displaystyle\int_{\Omega}u^{\frac{p_{0}}{p_{0}-1}(k+1-m)}\right)^{\frac{p_{0}-1}{p_{0}}}\left(\displaystyle\int_{\Omega}|\nabla
v|^{2p_{0}}\right)^{\frac{1}{p_{0}}}}\\\
&\leq\displaystyle{\frac{C_{3}\chi^{2}(k-1)}{2C_{D}}\|u^{\frac{k+m-1}{2}}\|^{\frac{2(k+1-m)}{k+m-1}}_{L^{\frac{2p_{0}(k+1-m)}{(p_{0}-1)(k+m-1)}}(\Omega)}}\\\
\end{array}$ (3.64)

by using (3.61), where $C_{3}>0$. Since, $k>|1-m|+q_{0}\frac{p_{0}-1}{p_{0}},$
we have

$\frac{q_{0}}{k+m-1}\leq{\frac{p_{0}(k+1-m)}{(p_{0}-1)(k+m-1)}}<+\infty,$

so that, the Gagliardo-Nirenberg inequality (Lemma 2.2) indicates that

$\begin{array}[]{rl}&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\|u^{\frac{k+m-1}{2}}\|^{\frac{2(k+1-m)}{k+m-1}}_{L^{\frac{2p_{0}(k+1-m)}{(p_{0}-1)(k+m-1)}}(\Omega)}}\\\
\leq&\displaystyle{C_{4}(\|\nabla
u^{\frac{k+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2(k+1-m)}{k+m-1}-\frac{2(p_{0}-1)q_{0}}{p_{0}(k+m-1)}}\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{2q_{0}}{k+m-1}}(\Omega)}^{\frac{2(p_{0}-1)q_{0}}{p_{0}(k+m-1)}}+\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{2q_{0}}{k+m-1}}(\Omega)}^{\frac{2(k+1-m)}{k+m-1}})}\\\
\leq&\displaystyle{C_{5}(\|\nabla
u^{\frac{k+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2(k+1-m)}{k+m-1}-\frac{2(p_{0}-1)q_{0}}{p_{0}(k+m-1)}}+1)}\\\
\end{array}$ (3.65)

with some positive constants $C_{4}$ as well as $C_{5}$, where $q_{0}$ is the
same as (3.60). Here we have used $L^{1}(\Omega)$ boundedness for $u$ (see
Lemma (3.1)). Due to (3.60), one has

${\frac{2(k+1-m)}{k+m-1}-\frac{2(p_{0}-1)q_{0}}{p_{0}(k+m-1)}}<2,$

so that, applying the Young inequality implies that there exists a positive
constant $C_{7}$ such that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\frac{\mu}{2}\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{C_{2}\int_{\Omega}u^{k}(v+1)+C_{7}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (3.66)

which together with the Young inequality implies that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{C_{9}\int_{\Omega}(v+1)^{k}+C_{8}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.67)

for some positive constants $C_{8}$ and $C_{9}.$ Now, in view of ${2p_{0}}>2$
(see (3.75))and $N=2$, then by Sobolev imbedding theorems, we derive from
(3.61) that

$\|v\|_{L^{\infty}(\Omega)}\leq C_{10}\|\nabla v\|_{L^{2p_{0}}(\Omega)},$

so that, combined with (3.67) implies that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{C_{11}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}),}\\\
\end{array}$ (3.68)

whereas a standard ODE comparison argument shows that (3.57) holds. ∎

######  Lemma 3.12.

Let $\Omega\subset\mathbb{R}^{N}(N\geq 1)$ be a bounded domain with smooth
boundary. Furthermore, assume that $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$,
where $\gamma_{*}$ is given by (3.56). If

${{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{2N}}}}>\frac{N}{2},$ (3.69)

then for any $k>1$, there exists a positive constant $C$ such that

$\|u(\cdot,t)\|_{L^{k}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (3.70)

where $\mu_{*}$ is given by (3.14).

###### Proof.

Due to

$m>\frac{2N}{N+{{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{N}}}}}.$

Now, in view of $\mu>0$ implies that

$\mu_{*}>1,$

which together with Lemma 3.10 results in

$m>1+\frac{[N-2{{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{2N}}}}]\mu_{*}}{2N\times{{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{2N}}}}}.$
(3.71)

Thus by (3.69), we may choose $q_{0,*}\in(1,\mu_{*})$ which is close to
$\mu_{*}$ such that

$\begin{array}[]{rl}m>&1+\frac{[N-2{{{\frac{(q_{0,*}+1)(N+q_{0,*}-1)}{2N}}}}]q_{0,*}}{2N\times{{{\frac{(q_{0,*}+1)(N+q_{0,*}-1)}{2N}}}}}\\\
\end{array}$ (3.72)

and

${{{\frac{(q_{0,*}+1)(N+q_{0,*}-1)}{2N}}}}>\frac{N}{2}.$ (3.73)

Next, observing that
$\frac{(q_{0,*}+1)(N+q_{0,*}-1)}{2N}\in(1,{{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{2N}}}})$,
thus, in light of Lemma 3.8, we derive that there exists a positive constant
$C_{1}$ such that

$\|\nabla v(\cdot,t)\|_{L^{2p_{0,*}}(\Omega)}\leq C_{1}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (3.74)

where

$p_{0,*}=\frac{(q_{0,*}+1)(N+q_{0,*}-1)}{2N}>\frac{N}{2}.$ (3.75)

Now, choosing
$k>\max\\{N+1,|1-m|+q_{0,*}\frac{p_{0,*}-1}{p_{0,*}},\frac{1-m}{2p_{0,*}-N}(2p_{0,*}N-N-2p_{0,*})\\}$
in (3.9), then, we have

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{2}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\frac{\mu}{2}\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{\frac{\chi^{2}(k-1)}{2m}\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}+C_{2}\int_{\Omega}u^{k}(v+1)~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.76)

with $C_{2}=\max\\{\xi\|w_{0}\|_{L^{\infty}(\Omega)},\mu+\kappa\xi\\}$.
According to the estimate of $\nabla v$ in (3.74) along with the Hölder
inequality we derive that there exists a positive constant $C_{3}>0$ such that

$\begin{array}[]{rl}&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\displaystyle\int_{\Omega}u^{k+1-m}|\nabla
v|^{2}}\\\
&\leq\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\left(\displaystyle\int_{\Omega}u^{\frac{p_{0,*}}{p_{0,*}-1}(k+1-m)}\right)^{\frac{p_{0,*}-1}{p_{0,*}}}\left(\displaystyle\int_{\Omega}|\nabla
v|^{2p_{0,*}}\right)^{\frac{1}{p_{0,*}}}}\\\
&\leq\displaystyle{\frac{C_{3}\chi^{2}(k-1)}{2C_{D}}\|u^{\frac{k+m-1}{2}}\|^{\frac{2(k+1-m)}{k+m-1}}_{L^{\frac{2p_{0,*}(k+1-m)}{(p_{0,*}-1)(k+m-1)}}(\Omega)}.}\\\
\end{array}$ (3.77)

In view of
$k>\max\\{|1-m|+q_{0,*}\frac{p_{0,*}-1}{p_{0,*}},\frac{1-m}{2p_{0,*}-N}(2p_{0,*}N-N-2p_{0,*})\\}$,
we have

$\frac{q_{0,*}}{k+m-1}\leq{\frac{p_{0,*}(k+1-m)}{(p_{0,*}-1)(k+m-1)}}<\frac{N}{N-2}.$

An application of the Gagliardo-Nirenberg inequality (see Lemma 2.2) implies
that for some positive constants $C_{4}$ as well as $C_{5}$ such that

$\begin{array}[]{rl}&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\|u^{\frac{k+m-1}{2}}\|^{\frac{2(k+1-m)}{k+m-1}}_{L^{\frac{2p_{0,*}(k+1-m)}{(p_{0,*}-1)(k+m-1)}}(\Omega)}}\\\
\leq&\displaystyle{C_{4}(\|\nabla
u^{\frac{k+m-1}{2}}\|_{L^{2}(\Omega)}^{2\frac{N(k+1-m-q_{0,*}+\frac{q_{0,*}}{p_{0,*}})}{N(k+m-1-q_{0,*})+2q_{0,*}}}\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{2q_{0,*}}{k+m-1}}(\Omega)}^{\frac{2(k+1-m)}{k+m-1}-2\frac{N(k+1-m-q_{0,*}+\frac{q_{0,*}}{p_{0,*}})}{N(k+m-1-q_{0,*})+2q_{0,*}}}+\|u^{\frac{k+m-1}{2}}\|_{L^{\frac{2q_{0,*}}{k+m-1}}(\Omega)}^{\frac{2(k+1-m)}{k+m-1}})}\\\
\leq&\displaystyle{C_{5}(\|\nabla
u^{\frac{k+m-1}{2}}\|_{L^{2}(\Omega)}^{2\frac{N(k+1-m-q_{0,*}+\frac{q_{0,*}}{p_{0,*}})}{N(k+m-1-q_{0,*})+2q_{0,*}}}+1)}\\\
\end{array}$ (3.78)

by using Lemma 3.1, where $q_{0,*}$ is the same as (3.72). Due to (3.72), one
has

${2\frac{N(k+1-m-q_{0,*}+\frac{q_{0,*}}{p_{0,*}})}{N(k+m-1-q_{0,*})+2q_{0,*}}}<2,$

so that, applying the Young inequality implies that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\frac{\mu}{2}\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{C_{2}\int_{\Omega}u^{k}(v+1)+C_{7}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),}\\\ \end{array}$ (3.79)

which together with the Young inequality again yields to

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{C_{9}\int_{\Omega}(v+1)^{k}+C_{8}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.80)

and some positive constants $C_{8}$ and $C_{9}.$ Now, in view of ${2p_{0}}>N$
(see (3.75)), then by Sobolev imbedding theorems, we derive from (3.74) that

$\|v\|_{L^{\infty}(\Omega)}\leq C_{10}\|\nabla v\|_{L^{2p_{0}}(\Omega)},$

so that, combined with (3.80) implies that

$\begin{array}[]{rl}&\displaystyle{\frac{1}{k}\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{(k-1)C_{D}}{4}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\mu\int_{\Omega}u^{k+1}}{}\\\
\leq&\displaystyle{C_{11}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}),}\\\
\end{array}$ (3.81)

whereas a standard ODE comparison argument shows that (3.70) holds.

∎

######  Lemma 3.13.

Let $\Omega\subset\mathbb{R}^{N}(N\neq 2)$ be a bounded domain with smooth
boundary. Furthermore, assume that $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$,
where $\gamma_{*}$ is given by (3.56). If

${{{\frac{(\mu_{*}+1)(N+\mu_{*}-1)}{2N}}}}\leq\frac{N}{2},$ (3.82)

then for any $k>1$, there exists a positive constant $C$ such that

$\|u(\cdot,t)\|_{L^{k}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (3.83)

###### Proof.

Let

$\bar{\beta}=\max\\{\frac{4N{{{\frac{(\mu_{*}+1)(N+\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}-1)}{N}}}}}{N+{{{\frac{(\mu_{*}+1)(N+\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}-1)}{N}}}}},2-\frac{2}{N}-m,16,8N+2\\}.$

Due to

$m>\frac{2N}{N+{{{\frac{(\mu_{*}+1)(N+\frac{\chi\max\\{1,\lambda_{0}\\}}{(\chi\max\\{1,\lambda_{0}\\}-\mu)_{+}}-1)}{N}}}}},$

as well as (3.82) and Lemma 3.8, we may choose $q_{0,***}\in(1,\mu_{*})$ which
is close to $\mu_{*}$ such that

$\begin{array}[]{rl}&\frac{2N(m-1)q_{0,***}}{N-2q_{0,***}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N}\\\
&>\displaystyle{\frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,***}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N}}\\\
\end{array}$ (3.84)

for all $\beta\geq\bar{\beta}.$ Therefore, we can choose

$k\in(\frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,***}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N},\frac{2N(m-1)q_{0,***}}{N-2q_{0,***}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N}).$
(3.85)

Now for the above $k$, by the Hölder inequality, we have

$\begin{array}[]{rl}J_{1}:&=\displaystyle{\displaystyle\frac{\chi^{2}(k-1)}{2C_{D}}\int_{\Omega}u^{{k}+1-m}|\nabla
v|^{2}}\\\
&\leq\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\left(\displaystyle\int_{\Omega}u^{{\frac{N}{N-2}}({k}+1-m)}\right)^{{\frac{N-2}{N}}}\left(\displaystyle\int_{\Omega}|\nabla
v|^{{N}}\right)^{{\frac{2}{N}}}}\\\
&=\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\|u^{\frac{{k}+m-1}{2}}\|^{\frac{2({k}+1-m)}{{k}+m-1}}_{L^{\frac{{\frac{2N}{N-2}}({k}+1-m)}{{k}+m-1}}(\Omega)}\|\nabla
v\|_{L^{{N}}(\Omega)}^{2}.}\\\ \end{array}$ (3.86)

Next, by (3.82), we derive that $m\geq 1$, so that, in view of $N\geq 3$, we
have

$\frac{1}{{k}+m-1}\leq\frac{{k}+1-m}{{k}+m-1}\frac{{\frac{N}{2}}}{({\frac{N}{2}}-1)_{+}}\leq\frac{N}{(N-2)_{+}},$

so that, by using Gagliardo-Nirenberg interpolation inequality (Lemma 2.2) and
$L^{1}(\Omega)$ boundedness for $u$ (see Lemma (3.1)) we have

$\begin{array}[]{rl}&\displaystyle{\frac{\chi^{2}(k-1)}{2C_{D}}\|u^{\frac{{k}+m-1}{2}}\|^{\frac{2({k}+1-m)}{{k}+m-1}}_{L^{\frac{{\frac{2N}{N-2}}({k}+1-m)}{{k}+m-1}}(\Omega)}}\\\
\leq&\displaystyle{C_{1}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\mu_{1}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{2}{{k}+m-1}}(\Omega)}^{1-\mu_{1}}+\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{2}{{k}+m-1}}(\Omega)})^{\frac{2({k}+1-m)}{{k}+m-1}}}\\\
\leq&\displaystyle{C_{2}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2({k}+1-m)\mu_{1}}{{k}+m-1}}+1)}\\\
\end{array}$ (3.87)

with some positive constants $C_{1}$ as well as $C_{2}$ and

$\mu_{1}=\frac{\frac{N[{k}+m-1]}{2}-\frac{N({k}+m-1)}{{\frac{2N}{N-2}}({k}+1-m)}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}=[{k}+m-1]\frac{\frac{N}{2}-\frac{N}{{\frac{2N}{N-2}}({k}+1-m)}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}\in(0,1).$

On the other hand, due to Lemma 2.2 and the fact that
$\beta\geq\bar{\beta}>\frac{N}{2}$ and $N>2$, we have

$\begin{array}[]{rl}\|\nabla
v\|_{L^{{N}}(\Omega)}^{2}=&\displaystyle{\||\nabla
v|^{\beta}\|_{L^{\frac{{N}}{\beta}}(\Omega)}^{\frac{2}{\beta}}}\\\
\leq&\displaystyle{C_{3}(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{2\mu_{2}}{\beta}}\||\nabla
v|^{\beta}\|_{L^{\frac{2q_{0,***}}{\beta}}(\Omega)}^{\frac{2(1-\mu_{2})}{\beta}}+\||\nabla
v|^{\beta}\|_{L^{\frac{2q_{0,***}}{\beta}}(\Omega)}^{\frac{2}{\beta}})}\\\
\leq&\displaystyle{C_{4}(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{2\mu_{2}}{\beta}}+1),}\\\ \end{array}$
(3.88)

where some positive constants $C_{3}$ as well as $C_{4}$ and

$\mu_{2}=\frac{\frac{N\beta}{2q_{0,***}}-\frac{N\beta}{{N}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,***}}}=\beta\frac{\frac{N}{2q_{0,***}}-\frac{N}{{N}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,***}}}\in(0,1).$

Inserting (3.87)–(3.88) into (3.86) as well as by means of the Young
inequality and (3.85) we see that for any $\delta>0,$

$\begin{array}[]{rl}J_{1}\leq&\displaystyle{C_{5}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2({k}+1-m)\mu_{1}}{{k}+m-1}}+1)(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{2\mu_{2}}{\beta}}+1)}\\\
=&\displaystyle{C_{5}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{{N}({{k}+1-m-{\frac{N-2}{N}}})}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}}+1)(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{{N}(\frac{2}{2q_{0,***}}-\frac{1}{{\frac{N}{2}}})}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,***}}}}+1)}\\\
\leq&\displaystyle{\delta\displaystyle\int_{\Omega}|\nabla
u^{\frac{m+k-1}{2}}|^{2}+\delta\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{2}+C_{6}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})}\\\ \end{array}$ (3.89)

for all $\beta\geq\bar{\beta}$. Here we have use the fact that
$k<\frac{2N(m-1)q_{0,***}}{N-2q_{0,***}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N}$
and $k>2-\frac{2}{N}-m.$ Next, due to the Hölder inequality and
$\beta\geq\bar{\beta}>16$, we have

$\begin{array}[]{rl}J_{2}:&=\displaystyle{\displaystyle\int_{\Omega}u^{2}|\nabla
v|^{2\beta-2}}\\\
&\leq\displaystyle{\left(\displaystyle\int_{\Omega}u^{{\frac{\beta}{4}}}\right)^{{\frac{8}{\beta}}}\left(\displaystyle\int_{\Omega}|\nabla
v|^{{\frac{\beta(2\beta-2)}{\beta-8}}}\right)^{{\frac{\beta-8}{\beta}}}}\\\
&\leq\displaystyle{\left(\displaystyle\int_{\Omega}u^{{\frac{\beta}{4}}}\right)^{{\frac{8}{\beta}}}\left(\displaystyle\int_{\Omega}|\nabla
v|^{{\frac{\beta(2\beta-2)}{\beta-8}}}\right)^{{\frac{\beta-8}{\beta}}}}\\\
&=\displaystyle{\|u^{\frac{{k}+m-1}{2}}\|^{\frac{4}{{k}+m-1}}_{L^{\frac{{\frac{\beta}{2}}}{{k}+m-1}}(\Omega)}\|\nabla
v\|_{L^{{\frac{\beta(2\beta-2)}{\beta-8}}}(\Omega)}^{(2\beta-2)}.}\\\
\end{array}$ (3.90)

On the other hand, with the help of $k>1-m+{\frac{N-2}{4N}}\beta$,
$\beta\geq\bar{\beta}\geq 4$ and Lemma 2.2 we conclude that

$\begin{array}[]{rl}&\displaystyle{\|u^{\frac{{k}+m-1}{2}}\|^{\frac{4}{{k}+m-1}}_{L^{\frac{{\frac{\beta}{2}}}{{k}+m-1}}(\Omega)}}\\\
\leq&\displaystyle{C_{7}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\mu_{3}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{2}{{k}+m-1}}(\Omega)}^{(1-\mu_{3})}+\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{2}{{k}+m-1}}(\Omega)})^{\frac{4}{{k}+m-1}}}\\\
\leq&\displaystyle{C_{8}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{4\mu_{3}}{{k}+m-1}}+1)}\\\
\end{array}$ (3.91)

with some positive constants $C_{7}$ as well as $C_{8}$ and

$\mu_{3}=\frac{\frac{N[{k}+m-1]}{2}-\frac{N({k}+m-1)}{{\frac{\beta}{2}}}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}=[{k}+m-1]\frac{\frac{N}{2}-\frac{N}{{\frac{\beta}{2}}}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}\in(0,1).$

On the other hand, again, it infers by the Gagliardo-Nirenberg inequality
(Lemma 2.2) that there are $C_{9}>0$ and $C_{10}>0$ ensuring

$\begin{array}[]{rl}\|\nabla
v\|_{L^{{\frac{\beta(2\beta-2)}{\beta-8}}}(\Omega)}^{(2\beta-2)}=&\displaystyle{\||\nabla
v|^{\beta}\|_{L^{\frac{{\frac{\beta(2\beta-2)}{\beta-8}}}{\beta}}(\Omega)}^{\frac{2\beta-2}{\beta}}}\\\
\leq&\displaystyle{C_{9}(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{(2\beta-2)\mu_{4}}{\beta}}\||\nabla
v|^{\beta}\|_{L^{\frac{2q_{0,***}}{\beta}}(\Omega)}^{\frac{(2\beta-2)(1-\mu_{4})}{\beta}}+\||\nabla
v|^{\beta}\|_{L^{\frac{2q_{0,***}}{\beta}}(\Omega)}^{\frac{(2\beta-2)}{\beta}})}\\\
\leq&\displaystyle{C_{10}(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{(2\beta-2)\mu_{4}}{\beta}}+1)}\\\
\end{array}$ (3.92)

for any $\beta\geq\bar{\beta}>\frac{7N+2}{2}$, where

$\mu_{4}=\frac{\frac{N\beta}{2q_{0,***}}-\frac{N\beta}{{\frac{\beta(2\beta-2)}{\beta-8}}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,***}}}=\beta\frac{\frac{N}{2q_{0,***}}-\frac{N}{{\frac{\beta(2\beta-2)}{\beta-8}}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,***}}}\in(0,1).$

Inserting (3.91)–(3.92) into (3.90) and using
$k>\frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,***}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N}$
and $k>2-\frac{2}{N}-m$ and Lemma 2.2, we derive that for the above
$\delta>0,$

$\begin{array}[]{rl}J_{2}\leq&\displaystyle{C_{11}(\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{{N}(2-{\frac{8}{\beta}})}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}}+1)(\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{\frac{{N}(\frac{2\beta-2}{2q_{0,***}}-{\frac{\beta-8}{\beta}})}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,***}}}}+1)}\\\
\leq&\displaystyle{\delta\displaystyle\int_{\Omega}|\nabla
u^{\frac{m+k-1}{2}}|^{2}+\delta\|\nabla|\nabla
v|^{\beta}\|_{L^{2}(\Omega)}^{2}+C_{12}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).}\\\ \end{array}$ (3.93)

Finally, with the help of (3.1) and by the Sobolev inequality, the Young
inequality and (3.85), we conclude that for the above $\delta>0,$ there exist
positive constants $C_{13},C_{14}$ as well as $C_{15}$ and $C_{16}$ such that

$\begin{array}[]{rl}J_{3}=&\displaystyle{\int_{\Omega}v^{k+1}}\\\
\leq&\displaystyle{C_{13}\|v\|^{k+1}_{L^{\infty}(\Omega)}}\\\
\leq&\displaystyle{C_{14}(\|\nabla v\|^{k+1}_{L^{N+1}(\Omega)}+1)}\\\
\leq&\displaystyle{C_{15}(\|\nabla v\|^{k+1}_{L^{2\beta}(\Omega)}+1)}\\\
\leq&\displaystyle{\delta\|\nabla v\|^{2\beta}_{L^{2\beta}(\Omega)}+C_{16}}\\\
\end{array}$ (3.94)

for all $\beta\geq\bar{\beta}>N+1.$ Now, inserting (3.89), (3.93)–(3.94) into
(3.48) and using the Young inequality and choosing $\delta$ small enough
yields to

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}(\frac{1}{k}\|u\|^{k}_{L^{k}(\Omega)}+\frac{1}{{2\beta}}\|\nabla
v\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)})+\frac{3(\beta-1)}{8{\beta^{2}}}\displaystyle\int_{\Omega}\left|\nabla|\nabla
v|^{\beta}\right|^{2}+\frac{\mu}{2}\int_{\Omega}u^{k+1}}\\\
&+\displaystyle{\frac{1}{2}\displaystyle\int_{\Omega}|\nabla
v|^{2\beta-2}|D^{2}v|^{2}+\displaystyle\frac{1}{2}\int_{\Omega}|\nabla
v|^{2\beta}+\frac{(k-1)C_{D}}{8}\int_{\Omega}u^{m+k-3}|\nabla u|^{2}}{}\\\
\leq&\displaystyle{C_{16}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max})}\\\
\end{array}$ (3.95)

and some positive constant $C_{16}$. Therefore, letting
$y:=\displaystyle\int_{\Omega}u^{k}+\displaystyle\int_{\Omega}|\nabla
v|^{2\beta}$ in (3.95) yields to

$\frac{d}{dt}y(t)+C_{17}y(t)\leq C_{18}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$

Thus a standard ODE comparison argument implies boundedness of $y(t)$ for all
$t\in(0,T_{max})$. Clearly, $\|u(\cdot,t)\|_{L^{k}(\Omega)}$ and $\|\nabla
v(\cdot,t)\|_{L^{2\beta}(\Omega)}$ are bounded for all $t\in(0,T_{max})$.
Obviously,
$\lim_{\beta\rightarrow+\infty}\displaystyle{\frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,***}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})}=\lim_{\beta\rightarrow+\infty}\frac{2N(m-1)q_{0,***}}{N-2q_{0,***}}(\frac{2}{N}-1+\frac{\beta}{q_{0,***}})+2-m-\frac{2}{N}=+\infty,$
hence, the boundedness of $\|u(\cdot,t)\|_{L^{k}(\Omega)}$ and the Hölder
inequality implies the results. ∎

Employing Lemmas 3.12–3.13, we can prove the following lemma.

######  Lemma 3.14.

Let $\Omega\subset\mathbb{R}^{N}(N\geq 1)$ be a bounded domain with smooth
boundary. Furthermore, assume that $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$,
where $\gamma_{*}$ is given by (3.56). Then for any $k>1$, there exists a
positive constant $C$ such that

$\|u(\cdot,t)\|_{L^{k}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (3.96)

Along with the Duhamel’s principle and $L^{p}$-$L^{q}$ estimates for Neumann
heat semigroup, the above lemma yields the following Lemma.

######  Lemma 3.15.

Let $(u,v,w)$ be the solution of the problem (1.3). Then

$\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max})$ (3.97)

###### Proof.

By Duhamel’s principle, we see that the solution $v$ can be expressed as
follows

$v(t)=e^{t(\Delta-1)}v_{0}+\int_{0}^{t}e^{(t-s)(\Delta-1)}v(s)ds,~{}t\in(0,T_{max}),$
(3.98)

where $(e^{t\Delta})_{t\geq 0}$ is the Neumann heat semigroup in $\Omega$.
Using Lemma 3.14, we follow the $L^{p}$-$L^{q}$ estimates for Neumann heat
semigroup to obtain for any $t\in(0,T_{max})$, we obtain

$\begin{array}[]{rl}&\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}\\\
\leq&\displaystyle{e^{-t}\|\nabla
v_{0}\|_{L^{\infty}(\Omega)}+\int_{0}^{t}(t-s)^{-\frac{N}{2}(\frac{1}{2N}-\frac{1}{2})}e^{-(t-s)}\|u(\cdot,s)\|_{L^{2N}(\Omega)}ds}\\\
\leq&\displaystyle{C_{1}~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}).}\\\
\end{array}$ (3.99)

This lemma is proved. ∎

Applying Lemma 3.14 and Lemma 3.15, a straightforward adaptation of the well-
established Moser-type iteration procedure [1] allows us to formulate a
general condition which is sufficient for the boundedness of $u$.

######  Lemma 3.16.

Let $(u,v,w)$ be the solution of the problem (1.3). Then there exists a
positive constant $C$ such that

$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}).$ (3.100)

###### Proof.

Firstly, by Lemma 3.14, we obtain that for any $k>0$,

$\|u(\cdot,t)\|_{L^{k}(\Omega)}\leq\alpha_{k}~{}~{}\mbox{for
all}~{}~{}t\in(0,T_{max}),$ (3.101)

where $\alpha_{k}$ depends on $k$. Multiplying the first equation of (1.3) by
$ku^{k-1}$ with $k\geq\max\\{2m,N+2\\}$, integrating it over $\Omega$, then
using the boundary condition $\frac{\partial u}{\partial\nu}=0$ and combining
with Lemma 3.15, we obtain

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+k(k-1)C_{D}\int_{\Omega}(u+\varepsilon)^{m-1}u^{k-2}|\nabla
u|^{2}{}+\mu k\int_{\Omega}u^{k+1}+\int_{\Omega}u^{k}}{}\\\
\leq&\displaystyle{k(k-1)\chi\int_{\Omega}u^{k-1}\nabla u\cdot\nabla
v-\xi\int_{\Omega}\nabla\cdot(u\nabla w)u^{k-1}+(\mu
k+1)\int_{\Omega}u^{k}}\\\
\leq&\displaystyle{k(k-1)\chi\int_{\Omega}u^{k-1}\nabla u\cdot\nabla v}\\\
&\displaystyle{+\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u^{k}vdx+\kappa\frac{({k-1})}{k}\xi\int_{\Omega}u^{k}dx+(\mu
k+1)\int_{\Omega}u^{k}}\\\
\leq&\displaystyle{\frac{k(k-1)C_{D}}{2}\int_{\Omega}(u+\varepsilon)^{m-1}u^{k-2}|\nabla
u|^{2}+C_{1}k\int_{\Omega}u^{k}+C_{2}k^{2}\int_{\Omega}u^{k}(u+\varepsilon)^{1-m},}\\\
\end{array}$ (3.102)

by using (3.12), where $C_{1}>0,C_{2}>0$, as all subsequently appearing
constants $C_{i}(i=3,4,\ldots)$ are independent of $k$. In what follows, we
estimate the last two terms of (3.102). When $m\geq 1$, then,

$(u+\varepsilon)^{1-m}\leq
u^{1-m}~{}~{}~{}\mbox{and}~{}~{}~{}(u+\varepsilon)^{m-1}\geq u^{m-1},$

so that, by (3.102), we have

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{k(k-1)C_{D}}{2}\int_{\Omega}u^{m+k-3}|\nabla
u|^{2}{}+\int_{\Omega}u^{k}}{}\\\
\leq&\displaystyle{C_{1}k\int_{\Omega}u^{k}+C_{2}k^{2}\int_{\Omega}u^{k+1-m}.}\\\
\end{array}$ (3.103)

In order to take full advantage of the dissipated quantities appearing on the
left-hand side herein, for any $\delta>0,$ we first invoke the Gagliardo-
Nirenberg inequality which provides $C_{3}>0,C_{4}>0$ as well as $C_{5}>0$ and
$C_{6}>0$ such that

$\begin{array}[]{rl}&\displaystyle{C_{1}k\|u\|^{k}_{L^{k}(\Omega)}}\\\
=&\displaystyle{C_{1}k\|u^{\frac{{k}+m-1}{2}}\|^{\frac{2k}{{k}+m-1}}_{L^{\frac{{2k}}{{k}+m-1}}(\Omega)}}\\\
\leq&\displaystyle{C_{1}k\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2Nk}{(N+2){k}+2N(m-1)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2k}{{k}+m-1}-\frac{2Nk}{(N+2){k}+2N(m-1)}}+C_{3}k\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2k}{{k}+m-1}}}\\\
\leq&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{2}+C_{4}k^{\frac{(N+2)k+2N(m-1)}{2k+2N(m-1)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{k[2k+N(m-1)]}{({k}+m-1)(k+N(m-1))}}+C_{3}k\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2k}{{k}+m-1}}}\\\
\leq&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{2}+C_{4}k^{\frac{N+2}{2}}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[2k+N(m-1)]}{(2k+2N(m-1))}}+C_{3}k\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k}}\\\
\end{array}$ (3.104)

and

$\begin{array}[]{rl}&\displaystyle{C_{2}k^{2}\|u\|^{k+1-m}_{L^{k+1-m}(\Omega)}}\\\
=&\displaystyle{C_{2}k^{2}\|u^{\frac{{k}+m-1}{2}}\|^{\frac{2(k+1-m)}{{k}+m-1}}_{L^{\frac{{2(k+1-m)}}{{k}+m-1}}(\Omega)}}\\\
\leq&\displaystyle{C_{5}k^{2}\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{\frac{2N[k-(2m-2)]}{(N+2){k}+2N(m-1)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2(k+1-m)}{{k}+m-1}-\frac{2N[k-(2m-2)]}{(N+2){k}+2N(m-1)}}+C_{5}k^{2}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2(k-1+m)}{{k}+m-1}}}\\\
\leq&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{2}+C_{6}k^{\frac{(N+2)k+2N(m-1)}{k+2N(m-1)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{k[k+2(m-1)]}{({k}+m-1)(\frac{k}{2}+N(m-1))}}+C_{5}k^{2}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2(k-1+m)}{{k}+m-1}}}\\\
\leq&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(\Omega)}^{2}+C_{6}k^{N+2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k+2(m-1)]}{(k+2N(m-1))}}+C_{5}k^{2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k+1-m}.}\\\
\end{array}$ (3.105)

Taking $\delta$ appropriately small, and substituting the above two
inequalities into (3.103), we obtain

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\int_{\Omega}u^{k}}{}\\\
\leq&\displaystyle{C_{7}k^{\frac{N+2}{2}}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[2k+N(m-1)]}{(2k+2N(m-1))}}+C_{8}k^{N+2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k+2(m-1)]}{(k+2N(m-1))}}+C_{9}k\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k}+C_{10}k^{2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k+1-m}.}\\\
\end{array}$ (3.106)

Notice that

${\frac{k[k+2(m-1)]}{(k+2N(m-1))}}\leq\max\\{\frac{k[2k+N(m-1)]}{(2k+2N(m-1))},k+1-m\\}\leq
k,$

we further obtain

$\begin{array}[]{rl}\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\int_{\Omega}u^{k}{}}\leq&\displaystyle{C_{11}k^{N+2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k+2(m-1)]}{(k+2N(m-1))}}+C_{12}k^{N+2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k}.}\\\
\end{array}$ (3.107)

In what follows, we use Moser iteration method to show the $L^{\infty}$
estimate of $u$. Take $k_{i}=2k_{i-1}=2^{i}k_{0},k_{0}=2m$,
$M_{i}=\sup_{t\in(0,T_{max})}\int_{\Omega}u^{\frac{{k_{i}}}{2}}$, then when
$m\geq 1$, we have

$\begin{array}[]{rl}\displaystyle{M_{i}}\leq&\displaystyle{\max\\{\lambda^{i}M_{i-1}^{2},\|1+u_{0}\|_{L^{\infty}(\Omega)}^{k_{i}}\\}.}\\\
\end{array}$ (3.108)

with some $\lambda>1.$ Now if
$\lambda^{i}M_{i-1}^{2}\leq\|1+u_{0}\|_{L^{\infty}(\Omega)}^{k_{i}}$ for
infinitely many $i\geq 1$, we get (3.33) with
$C=\|1+u_{0}\|_{L^{\infty}(\Omega)}$. Otherwise
$M_{i}\leq\lambda^{i}M_{i-1}^{2}$ for all $i=0,1,\ldots$, so $\ln M_{i}\leq
i\ln\lambda+2\ln M_{i-1}$. By induction, we get

$\ln M_{i}\leq(i+2)\ln\lambda+2^{i}(\ln M_{0}+2\ln\lambda)$

and thus

$M_{i}\leq\lambda^{i+2+2^{i}}M_{0}^{2^{i}}.$

From this, it follows that (3.100) is valid with some positive constant. When
$0<m<1$, noticing that $u^{k}(u+\varepsilon)^{1-m}\leq
u^{k+1-m}+\varepsilon^{1-m}u^{k}$, then by (3.101) and $\varepsilon\in(0,1)$,
we derive from Lemma 3.15 that

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+\frac{k(k-1)C_{D}}{2^{2-m}}\int_{u>\varepsilon}u^{m+k-3}|\nabla
u|^{2}{}+\mu k\int_{\Omega}u^{k+1}+\mu k\int_{\Omega}u^{k}}{}\\\
\leq&\displaystyle{C_{13}k^{2}\int_{\Omega}u^{k}+C_{14}k^{2}\int_{\Omega}u^{k+1-m}}\\\
\leq&\displaystyle{C_{13}k^{2}\int_{u\leq\varepsilon}u^{k}+C_{13}k^{2}\int_{u>\varepsilon}u^{k}+C_{14}k^{2}\int_{u\leq\varepsilon}u^{k+1-m}+C_{14}k^{2}\int_{u>\varepsilon}u^{k+1-m}}\\\
\leq&\displaystyle{C_{13}k^{2}\int_{u\leq\varepsilon}u^{\frac{k}{2}}+C_{14}k^{2}\int_{u\geq\varepsilon}u^{k}+C_{14}k^{2}\int_{u\geq\varepsilon}u^{k+1-m}.}\\\
\end{array}$ (3.109)

Denote $J(t)=\\{x\in u>\varepsilon\\}$. By virtue of the Gagliardo-Nirenberg
interpolation inequality and (3.101), we obtain

$\|u\|^{k}_{L^{k}(J(t))}\leq\|u\|_{L^{3}(J(t))}\|u\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}\leq
C_{15}\|u\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}$

and

$\|u\|^{k+1-m}_{L^{k+1-m}(J(t))}\leq\|u\|_{L^{3(2-m)}(J(t))}^{2-m}\|u\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}\leq
C_{16}\|u\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}.$

Substituting the above two inequalities into (3.102), we obtain

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+k(k-1)C_{D}\int_{J(t)}(u+\varepsilon)^{m-1}u^{k-2}|\nabla
u|^{2}{}+\mu k\int_{\Omega}u^{k+1}+\int_{\Omega}u^{k}}{}\\\
\leq&\displaystyle{C_{17}k^{2}\int_{\Omega}u^{\frac{k}{2}}+C_{18}k^{2}\|u\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}.}\\\
\end{array}$ (3.110)

Using the Gagliardo-Nirenberg interpolation inequality again, it follows

$\begin{array}[]{rl}&C_{18}k^{2}\|u\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}\\\
=&C_{18}k^{2}\|u^{\frac{{k}+m-1}{2}}\|^{\frac{2(k-1)}{m+k-1}}_{L^{\frac{3(k-1)}{k+m-1}}(J(t))}\\\
\leq&\displaystyle{C_{19}k^{2}\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(J(t))}^{\frac{4(2k-N)}{(N+2){k}+2N(m-1)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2(k-1)}{m+k-1}-\frac{4(2k-N)}{(N+2){k}+2N(m-1)}}+C_{19}k^{2}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2(k-1)}{{k}+m-1}}}\\\
\leq&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(J(t))}^{2}+C_{20}k^{\frac{2[(N+2){k}+2N(m-1)]}{(N-2)k+2Nm}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2k[k(N-2)+(2N-4)m+2-N]}{(m+k-1)[(N-2){k}+2Nm]}}+C_{19}k^{2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k-1}}\\\
\leq&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(J(t))}^{2}+C_{20}k^{\frac{2(N+2)}{(N-2)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{{k}+m-1}}(\Omega)}^{\frac{2k[k(N-2)+(2N-4)m+2-N]}{(m+k-1)[(N-2){k}+2Nm]}}+C_{19}k^{2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k-1}}\\\
=&\displaystyle{\delta\|\nabla
u^{\frac{{k}+m-1}{2}}\|_{L^{2}(J(t))}^{2}+C_{20}k^{\frac{2(N+2)}{(N-2)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k(N-2)+(2N-4)m+2-N]}{[(N-2){k}+2Nm]}}+C_{19}k^{2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k-1},}\\\
\end{array}$

substituting this inequality into (3.110) gives

$\begin{array}[]{rl}&\displaystyle{\frac{d}{dt}\|u\|^{k}_{L^{k}(\Omega)}+k(k-1)C_{D}\int_{J(t)}(u+\varepsilon)^{m-1}u^{k-2}|\nabla
u|^{2}{}+\mu k\int_{\Omega}u^{k+1}+\int_{\Omega}u^{k}}{}\\\
\leq&\displaystyle{C_{21}k^{2}\int_{\Omega}u^{\frac{k}{2}}+C_{23}k^{\frac{2(N+2)}{(N-2)}}\|u^{\frac{{k}+m-1}{2}}\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k(N-2)+(2N-4)m+2-N]}{[(N-2){k}+2Nm]}}+C_{22}k^{2}\|u\|_{L^{\frac{k}{2}}(\Omega)}^{k-1}.}\\\
\end{array}$ (3.111)

Similarly to the case $m\geq 1$, and we obtain (3.100).

∎

By virtue of (2.1) and Lemmas 3.15–3.16, we are now in a position to prove
Theorem 1.1.

Proof of Theorem 1.1.

###### Proof.

From (2.1) and Lemmas 3.15–3.16, we derive that there exists a constant $C>0$
independent of $\varepsilon$ such that

$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,T_{max}).$ (3.112)

Suppose on the contrary that $T_{max}<\infty$, then (4.8) contradicts to the
blow-up criterion (2.1), which implies $T_{max}=\infty$. Therefore, the
classical solution $(u,v,w)$ is global in time and bounded. ∎

##  4 Proof of Theorem 1.2

The goal of this section is to prove theorem 1.2. In the absence of (1.5), the
first equation in system (1.3) may be degenerate, so system (1.3) might not
have classical solutions. Our goal is to construct solutions of (1.3) as
limits of solutions to appropriately regularized problems. To this end, we
approximate the diffusion coefficient function in (1.3) by a family
$(D_{\varepsilon})_{\varepsilon\in(0,1)}$ of functions

$D_{\varepsilon}\in C^{2}((0,\infty))~{}\mbox{such
that}~{}~{}D_{\varepsilon}(u)\geq\varepsilon~{}\mbox{for all}~{}u>0$
$~{}~{}\mbox{and}~{}D(u)\leq D_{\varepsilon}(u)\leq
D(u)+2\varepsilon~{}\mbox{for all}~{}u>0~{}\mbox{and}~{}\varepsilon\in(0,1).$

Therefore, for any $\varepsilon\in(0,1)$, the regularized problem of (1.3) is
presented as follows

$\left\\{\begin{array}[]{ll}u_{\varepsilon
t}=\nabla\cdot(D_{\varepsilon}(u_{\varepsilon})\nabla
u_{\varepsilon})-\chi\nabla\cdot(u_{\varepsilon}\nabla
v_{\varepsilon})-\xi\nabla\cdot(u_{\varepsilon}\nabla w_{\varepsilon})+\mu
u_{\varepsilon}(1-u_{\varepsilon}-w_{\varepsilon}),\quad x\in\Omega,t>0,\\\
\displaystyle{v_{\varepsilon t}=\Delta
v_{\varepsilon}+u_{\varepsilon}-v_{\varepsilon}},\quad x\in\Omega,t>0,\\\
\displaystyle{w_{\varepsilon t}=-v_{\varepsilon}w_{\varepsilon}},\quad
x\in\Omega,t>0,\\\ \displaystyle{\frac{\partial
u_{\varepsilon}}{\partial\nu}=\frac{\partial
v_{\varepsilon}}{\partial\nu}=\frac{\partial
w_{\varepsilon}}{\partial\nu}=0},\quad x\in\partial\Omega,t>0,\\\
\displaystyle{u_{\varepsilon}(x,0)=u_{0}(x)},v_{\varepsilon}(x,0)=v_{0}(x),w_{\varepsilon}(x,0)=w_{0}(x),\quad
x\in\Omega.\\\ \end{array}\right.$ (4.1)

We are now in the position to construct global weak solutions for (1.3).
Before going into details, let us first give the definition of weak solution.

######  Definition 4.1.

Let $T\in(0,\infty]$, and $\Omega\subset R^{N}$ be a bounded domain with
smooth boundary. A triple $(u,v,w)$ of nonnegative functions defined on
$\Omega\times(0,T)$, is called a weak solution to (1.3) on $[0,T)$ if

$\begin{array}[]{ll}\mbox{(i)}~{}~{}u\in
L_{loc}^{2}([0,T);L^{2}(\Omega)),~{}~{}v\in
L_{loc}^{2}([0,T);W^{1,2}(\Omega)),~{}~{}w\in
L_{loc}^{2}([0,T);W^{1,2}(\Omega));\\\ \end{array}$ (4.2)
$\begin{array}[]{rl}\mbox{(ii)}~{}~{}H(u)\in
L^{1}_{loc}(\bar{\Omega}\times[0,T)),~{}~{}u\nabla
v~{}~{}\mbox{and}~{}~{}u\nabla w~{}~{}\in
L^{1}_{loc}(\bar{\Omega}\times[0,T);\mathbb{R}^{N});\end{array}$ (4.3)

(iii) $(u,v,w)$ satisfies (1.3) in the sense that for every $\varphi\in
C_{0}^{\infty}(\bar{\Omega}\times[0,T))$

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{T}\int_{\Omega}u\varphi_{t}-\int_{\Omega}u_{0}\varphi(\cdot,0)}\\\
=&\displaystyle{\int_{0}^{T}\int_{\Omega}H(u)\Delta\varphi+\chi\int_{0}^{T}\int_{\Omega}u\nabla
v\cdot\nabla\varphi+\xi\int_{0}^{T}\int_{\Omega}u\nabla
w\cdot\nabla\varphi}+\displaystyle{\int_{0}^{T}\int_{\Omega}(\mu u-uw-\mu
u^{2})\varphi;}\end{array}$ (4.4)

holds as well as

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{T}\int_{\Omega}v\varphi_{t}-\int_{\Omega}v_{0}\varphi(\cdot,0)=-\int_{0}^{T}\int_{\Omega}\nabla
v\cdot\nabla\varphi-\int_{0}^{T}\int_{\Omega}v\varphi+\int_{0}^{T}\int_{\Omega}u\varphi}\end{array}$
(4.5)

and

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{T}\int_{\Omega}w\varphi_{t}-\int_{\Omega}w_{0}\varphi(\cdot,0)=-\int_{0}^{T}\int_{\Omega}vw\varphi,}\end{array}$
(4.6)

where we let

$H(s)=\int_{0}^{s}D(\sigma)d\sigma~{}~{}\mbox{for}~{}~{}s\geq 0.$ (4.7)

In particular, if $T=\infty$ can be taken, then $(u,v,w)$ is called a global-
in-time weak solution to (1.3).

We proceed to establish the main step towards the boundedness of weak
solutions to (1.3). To this end, firstly, from (2.1) and Lemmas 3.15–3.16, we
can easily derive the following estimates for $u_{\varepsilon}$ and
$v_{\varepsilon}$, which plays an important role in proving Theorem 1.2.

######  Lemma 4.1.

Let $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$, where $\gamma_{*}$ is given by
(3.56). Then one can find $C>0$ independent of $\varepsilon\in(0,1)$ such that

$\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,\infty)$ (4.8)

and

$\|v_{\varepsilon}(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C~{}~{}\mbox{for
all}~{}~{}t\in(0,\infty).$ (4.9)

Now with the above boundedness information at hand, we may invoke standard
parabolic regularity to obtain the Hölder regularity properties.

######  Lemma 4.2.

Let $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$, where $\gamma_{*}$ is given by
(3.56). Then for any $\varepsilon\in(0,1)$, one can find $\mu\in(0,1)$ such
that for some $C>0$

$\|v_{\varepsilon}(\cdot,t)\|_{C^{\mu,\frac{\mu}{2}}(\Omega\times[t,t+1])}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(0,\infty),$ (4.10)

and such that for any $\tau>0$ there exists $C(\tau)>0$ fulfilling

$\|\nabla
v_{\varepsilon}(\cdot,t)\|_{C^{\mu,\frac{\mu}{2}}(\Omega\times[t,t+1])}\leq
C~{}~{}\mbox{for all}~{}~{}t\in(\tau,\infty).$ (4.11)

###### Proof.

Firstly, by Lemma 4.1, we derive that $g_{\varepsilon}$ is bounded in
$L^{\infty}(\Omega\times(0,\infty))$, where
$g_{\varepsilon}(x,t):=-v_{\varepsilon}(x,t)+u_{\varepsilon}(x,t)$ for all
$(x,t)\in\Omega\times(0,\infty)$. Therefore, in view of the standard parabolic
regularity theory to the second equation of (4.1), one has (4.10) and (4.11)
hold. ∎

To achieve the convergence result, we need to derive some regularity
properties of time derivatives.

######  Lemma 4.3.

Let $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$, where $\gamma_{*}$ is given by
(3.56). Moreover, let $\varsigma>m$ and $\varsigma\geq 2(m-1)$. Then for all
$T>0$ and $\varepsilon\in(0,1)$, there exists $C(T)>0$ such that

$\int_{0}^{T}\|\partial_{t}u_{\varepsilon}^{\varsigma}(\cdot,t)\|_{(W^{2,q}(\Omega))^{*}}dt\leq
C(T)$ (4.12)

and

$\|w_{\varepsilon}(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C(T).$ (4.13)

###### Proof.

Firstly, with the help of Lemma 4.1, for all $\varepsilon\in(0,1),$ we can fix
a positive constants $C_{1}$ such that

$u_{\varepsilon}\leq C_{1}~{}~{}\mbox{and}~{}~{}|\nabla v_{\varepsilon}|\leq
C_{1}~{}~{}\mbox{in}~{}~{}\Omega\times(0,\infty).$ (4.14)

Next, we will prove (4.12). To this end, for any fixed $\psi\in
C_{0}^{\infty}(\Omega)$, multiplying the first equation by
$u^{\varsigma-1}_{\varepsilon}\psi$, we have

$\begin{array}[]{rl}&\displaystyle\frac{1}{\varsigma}\int_{\Omega}\partial_{t}u^{\varsigma}_{\varepsilon}(\cdot,t)\cdot\psi\\\
=&\displaystyle{\int_{\Omega}u^{\varsigma-1}_{\varepsilon}\left[\nabla\cdot(D_{\varepsilon}(u_{\varepsilon})\nabla
u_{\varepsilon})-\chi\nabla\cdot(u_{\varepsilon}\nabla
v_{\varepsilon})-\xi\nabla\cdot(u_{\varepsilon}\nabla w_{\varepsilon})+\mu
u_{\varepsilon}(1-w_{\varepsilon}-u_{\varepsilon})\right]\cdot\psi}\\\
=&\displaystyle{-(\varsigma-1)\int_{\Omega}u_{\varepsilon}^{\varsigma-2}D_{\varepsilon}(u_{\varepsilon})|{\nabla}{n}_{\varepsilon}|^{2}\psi-\int_{\Omega}u_{\varepsilon}^{\varsigma-1}D_{\varepsilon}(u_{\varepsilon}){\nabla}{u}_{\varepsilon}\cdot\nabla\psi}\\\
&+\displaystyle{(\varsigma-1)\chi\int_{\Omega}u_{\varepsilon}^{\varsigma-1}\nabla
u_{\varepsilon}\cdot\nabla
v_{\varepsilon}\psi+\chi\int_{\Omega}u_{\varepsilon}^{\varsigma}\nabla
v_{\varepsilon}\cdot\nabla\psi}\\\
&+\displaystyle{(\varsigma-1)\xi\int_{\Omega}u_{\varepsilon}^{\varsigma-1}\nabla
u_{\varepsilon}\cdot\nabla
w_{\varepsilon}\psi+\xi\int_{\Omega}u_{\varepsilon}^{\varsigma}\nabla
w_{\varepsilon}\cdot\nabla\psi}\\\
&+\displaystyle{\mu\int_{\Omega}u_{\varepsilon}^{\varsigma}\psi-\mu\int_{\Omega}u_{\varepsilon}^{\varsigma}w_{\varepsilon}\psi-\mu\int_{\Omega}u_{\varepsilon}^{\varsigma+1}\psi~{}~{}\mbox{for
all}~{}~{}t\in(0,\infty).}\\\ \end{array}$ (4.15)

Next, we will estimate the right-hand sides of (4.15). To this end, assuming
that $p:=\varsigma-m+1$, then $\varsigma>m$ and $\varsigma\geq 2(m-1)$ yield
to $p>1$ and $p\geq m-1$. Since (4.14), we integrate (3.9) with respect to $t$
over $(0,T)$ for some fixed $T>0$ and then have

$\begin{array}[]{rl}&\displaystyle{\frac{1}{{p}}\int_{\Omega}u_{\varepsilon}^{{{p}}}(\cdot,T)+\frac{C_{D}(p-1)}{2}\int_{0}^{T}\int_{\Omega}u_{\varepsilon}^{m+p-3}|\nabla
u_{\varepsilon}|^{2}}\\\
\leq&\displaystyle{\frac{(p-1)\chi^{2}}{2C_{D}}\int_{0}^{T}\int_{\Omega}u_{\varepsilon}^{p+1-m}|\nabla
v_{\varepsilon}|^{2}+\frac{({k-1})}{k}\xi\|w_{0}\|_{L^{\infty}(\Omega)}\int_{\Omega}u_{\varepsilon}^{k}v_{\varepsilon}}\\\
&\displaystyle{+(\mu+\kappa\xi)\int_{\Omega}u_{\varepsilon}^{k}+\mu
C_{1}^{p}T+\frac{1}{{p}}\int_{\Omega}n_{0}^{{{p}}}}\\\
\leq&\displaystyle{\frac{(p-1)\chi^{2}}{2C_{D}}C_{1}^{p+3-m}T+\mu
C_{1}^{p}T+\frac{1}{{p}}\int_{\Omega}n_{0}^{{{p}}}.}\\\ \end{array}$ (4.16)

On the other hand, by $p=\varsigma-m+1$, we have

$\int_{0}^{T}\int_{\Omega}u_{\varepsilon}^{\varsigma-2}|\nabla
u_{\varepsilon}|^{2}=\int_{0}^{T}\int_{\Omega}u_{\varepsilon}^{m+p-3}|\nabla
u_{\varepsilon}|^{2}\leq C_{2}(1+T)$ (4.17)

for some positive constant $C_{2}$. Next, by (4.14), we also derive that

$\begin{array}[]{rl}&\displaystyle{\mu\int_{\Omega}u_{\varepsilon}^{\varsigma}\psi-\mu\int_{\Omega}u_{\varepsilon}^{\varsigma}w_{\varepsilon}\psi-\mu\int_{\Omega}u_{\varepsilon}^{\varsigma+1}\psi\leq
C_{1}^{\varsigma}|\Omega|(\mu+\mu\|w_{0}\|_{L^{\infty}(\Omega)}+\mu
C_{1})\|\psi\|_{L^{\infty}(\Omega)}}\\\ \end{array}$ (4.18)

for all $\varepsilon\in(0,1).$ Moreover, by (4.14)–(4.18) and the Young
inequality, we conclude that there exists $C_{3}>0$ such that

$\begin{array}[]{rl}\displaystyle{|\int_{\Omega}\partial_{t}u^{\varsigma}_{\varepsilon}(\cdot,t)\cdot\psi|\leq
C_{3}(\int_{\Omega}u_{\varepsilon}^{\varsigma-2}|\nabla
u_{\varepsilon}|^{2}+1)\|\psi\|_{W^{1,\infty}(\Omega)}.}\\\ \end{array}$
(4.19)

Due to the embedding $W^{2,q}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)$ for
$q>N$, we deduce that there exists $C_{4}>0$ such that

$\begin{array}[]{rl}\displaystyle{\|\partial_{t}u_{\varepsilon}^{\varsigma}(\cdot,t)\|_{(W^{2,q}(\Omega))^{*}}\leq
C_{4}(\int_{\Omega}u_{\varepsilon}^{\varsigma-2}|\nabla
u_{\varepsilon}|^{2}+1)~{}~{}\mbox{for all}~{}~{}t\in(0,\infty)~{}~{}\mbox{and
any}~{}~{}\varepsilon\in(0,1).}\\\ \end{array}$ (4.20)

Now, combining (4.17) and (4.20), we can get (4.12). Now, observing that the
third equation of (4.1) is an ODE, we derive that for any
$(x,t)\in\Omega\times(0,\infty),$

$\begin{array}[]{rl}&\displaystyle{w_{\varepsilon}(x,t)=w_{0}(x)e^{-\int_{0}^{t}v_{\varepsilon}(x,s)ds}.}\\\
\end{array}$ (4.21)

Hence, by a basic calculation, we conclude that for any
$(x,t)\in\Omega\times(0,\infty),$

$\begin{array}[]{rl}&\displaystyle{\nabla w_{\varepsilon}(x,t)=\nabla
w_{0}(x)e^{-\int_{0}^{t}v_{\varepsilon}(x,s)ds}-w_{0}(x)e^{-\int_{0}^{t}v_{\varepsilon}(x,s)ds}\int_{0}^{t}\nabla
v_{\varepsilon}(x,s)ds,}\\\ \end{array}$ (4.22)

which together with (1.6) and (4.14) implies (4.13). ∎

We are now in the position to prove our main result on global weak
solvability.

######  Lemma 4.4.

Assume that $m>\frac{2N}{N+\gamma_{*}}$ with $N\geq 1$, where $\gamma_{*}$ is
given by (3.56). Then there exists
$(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ such that
$\varepsilon_{j}\rightarrow 0$ as $j\rightarrow\infty$ and that

$u_{\varepsilon}\rightarrow
u~{}~{}\mbox{a.e.}~{}~{}\mbox{in}~{}~{}\Omega\times(0,\infty),$ (4.23)
$u_{\varepsilon}\rightharpoonup u~{}~{}\mbox{weakly star
in}~{}~{}L^{\infty}(\Omega\times(0,\infty)),$ (4.24)
$v_{\varepsilon}\rightarrow
v~{}~{}\mbox{in}~{}~{}C^{0}_{loc}(\bar{\Omega}\times[0,\infty)),$ (4.25)
$\nabla v_{\varepsilon}\rightarrow\nabla
v~{}~{}\mbox{in}~{}~{}C^{0}_{loc}(\bar{\Omega}\times[0,\infty)),$ (4.26)
$\nabla v_{\varepsilon}\rightarrow\nabla
v~{}~{}\mbox{in}~{}~{}L^{\infty}(\Omega\times(0,\infty))$ (4.27)

as well as

$w_{\varepsilon}\rightharpoonup w~{}~{}\mbox{weakly star
in}~{}~{}L^{\infty}(\Omega\times(0,\infty))$ (4.28)

and

$\nabla w_{\varepsilon}\rightharpoonup\nabla w~{}~{}\mbox{weakly star
in}~{}~{}L^{\infty}_{loc}(\Omega\times(0,\infty))$ (4.29)

with some triple $(u,v,w)$ which is a global weak solution of (4.1) in the
sense of Definition 4.1.

###### Proof.

Firstly, due to Lemma 4.1 and Lemma 4.3, for each $T>0$, we can find
$\varepsilon$-independent constant $C(T)$ such that for all $t\in(0,T)$,

$\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v_{\varepsilon}(\cdot,t)\|_{W^{1,\infty}(\Omega)}+\|w_{\varepsilon}(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq
C(T)~{}~{}$ (4.30)

as well as

$\int_{0}^{T}\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-3}|\nabla
u_{\varepsilon}|^{2}\leq C(T)~{}~{}~{}\mbox{for
any}~{}~{}p>1~{}~{}~{}\mbox{and}~{}~{}p\geq m-1.$ (4.31)

Now, choosing $\varphi\in W^{1,2}(\Omega)$ as a second function in the first
equation in (4.1) and using (4.30), we have

$\begin{array}[]{rl}&\displaystyle\left|\int_{\Omega}(v_{\varepsilon,t})\varphi\right|\\\
=&\displaystyle{\left|\int_{\Omega}\left[\Delta
v_{\varepsilon}-v_{\varepsilon}+u_{\varepsilon}\right]\varphi\right|}\\\
=&\displaystyle{\left|\int_{\Omega}\left[-\nabla
v_{\varepsilon}\cdot\nabla\varphi+v_{\varepsilon}\varphi+u_{\varepsilon}\varphi\right]\right|}\\\
\leq&\displaystyle{\left\\{\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega)}+\|v_{\varepsilon}\|_{L^{2}(\Omega)}+\|u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}}\times\displaystyle{\|\varphi\|_{W^{1,2}(\Omega)}}\\\
\end{array}$

for all $t>0$. Along with (4.30), further implies that

$\begin{array}[]{rl}&\displaystyle\int_{0}^{T}\|\partial_{t}v_{\varepsilon}(\cdot,t)\|_{({W^{1,2}(\Omega)})^{*}}^{2}dt\\\
\leq&\displaystyle{\int_{0}^{T}\left\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\|v_{\varepsilon}\|_{L^{2}(\Omega)}+\|n_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}}^{2}dt\\\
\leq&\displaystyle{C_{1}(T),}\\\ \end{array}$ (4.32)

where $C_{1}$ is a positive constant independent of $\varepsilon$. Combining
estimates (4.30)–(4.32) and the fact that
$w_{\varepsilon}\leq\|w_{0}\|_{L^{\infty}(\Omega)}$, we can pick a sequence
$(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ with
$\varepsilon=\varepsilon_{j}\searrow 0$ as $j\rightarrow\infty$ such that
(4.24)–(4.29) are valid with certain limit functions $u,v$ and $w$ belonging
to the indicated spaces. We next fix $\varsigma>m$ satisfying $\varsigma\geq
2(m-1)$ and set $p:=2\zeta-m+1$, then by (4.31) implies that for each $T>0,$
$(u_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}$ is bounded in
$L^{2}((0,T);W^{1,2}(\Omega))$. With the help of Lemma 4.3, we also show that

$(\partial_{t}u_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}~{}~{}\mbox{is
bounded in}~{}~{}L^{1}((0,T);(W^{2,q}(\Omega))^{*})~{}~{}\mbox{for
each}~{}~{}T>0$

and some $q>N.$ Hence, an Aubin-Lions lemma (see e.g. [36]) applies to the
above inequality we have the strong precompactness of
$(u_{\varepsilon}^{\varsigma})_{\varepsilon\in(0,1)}$ in
$L^{2}(\Omega\times(0,T))$. Therefore, we can pick a suitable subsequence such
that $u_{\varepsilon}^{\varsigma}\rightarrow z^{\varsigma}$ for some
nonnegative measurable $z:\Omega\times(0,\Omega)\rightarrow\mathbb{R}$. In
light of (4.24) and the Egorov theorem, we have $z=u$ necessarily, so that
(4.23) is valid.

Next we shall prove that $(u,v,w)$ is a weak solution of problem (1.3). To
this end, multiplying the first equation as well as the second equation and
third equation in (4.1) by $\varphi\in
C^{\infty}_{0}(\Omega\times[0,\infty))$, we obtain

$\begin{array}[]{rl}\displaystyle{-\int_{0}^{\infty}\int_{\Omega}u_{\varepsilon}\varphi_{t}-\int_{\Omega}u_{0}\varphi(\cdot,0)}=&\displaystyle{\int_{0}^{\infty}\int_{\Omega}H(u_{\varepsilon}+\varepsilon)ds\Delta\varphi+\chi\int_{0}^{\infty}\int_{\Omega}u_{\varepsilon}\nabla
v_{\varepsilon}\cdot\nabla\varphi}\\\
&+\displaystyle{\xi\int_{0}^{\infty}\int_{\Omega}u_{\varepsilon}\nabla
w_{\varepsilon}\cdot\nabla\varphi+\int_{0}^{\infty}\int_{\Omega}(\mu
u_{\varepsilon}-\mu u_{\varepsilon}w_{\varepsilon}-\mu
u_{\varepsilon}^{2})\varphi}\\\ \end{array}$ (4.33)

as well as

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{\infty}\int_{\Omega}v_{\varepsilon}\varphi_{t}-\int_{\Omega}v_{0}\varphi(\cdot,0)=-\int_{0}^{\infty}\int_{\Omega}\nabla
v_{\varepsilon}\cdot\nabla\varphi-\int_{0}^{\infty}\int_{\Omega}v_{\varepsilon}\varphi+\int_{0}^{\infty}\int_{\Omega}u_{\varepsilon}\varphi}\end{array}$
(4.34)

and

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{\infty}\int_{\Omega}w_{\varepsilon}\varphi_{t}-\int_{\Omega}w_{0}\varphi(\cdot,0)=-\int_{0}^{\infty}\int_{\Omega}v_{\varepsilon}w_{\varepsilon}\varphi.}\end{array}$
(4.35)

for all $\varepsilon\in(0,1)$, where $H$ is given by (4.7). Then
(4.23)–(4.27), and the dominated convergence theorem enables us to conclude

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{\infty}\int_{\Omega}u\varphi_{t}-\int_{\Omega}u_{0}\varphi(\cdot,0)}\\\
=&\displaystyle{\int_{0}^{\infty}\int_{\Omega}H(u)\Delta\varphi+\chi\int_{0}^{\infty}\int_{\Omega}u\nabla
v\cdot\nabla\varphi+\xi\int_{0}^{\infty}\int_{\Omega}u\nabla
w\cdot\nabla\varphi}+\displaystyle{\int_{0}^{\infty}\int_{\Omega}(\mu u-\mu
uw-\mu u^{2})\varphi}\end{array}$

as well as

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{\infty}\int_{\Omega}v\varphi_{t}-\int_{\Omega}v_{0}\varphi(\cdot,0)=-\int_{0}^{\infty}\int_{\Omega}\nabla
v\cdot\nabla\varphi-\int_{0}^{\infty}\int_{\Omega}v\varphi+\int_{0}^{\infty}\int_{\Omega}u\varphi}\end{array}$

and

$\begin{array}[]{rl}&\displaystyle{-\int_{0}^{\infty}\int_{\Omega}w\varphi_{t}-\int_{\Omega}w_{0}\varphi(\cdot,0)=-\int_{0}^{\infty}\int_{\Omega}vw\varphi}\end{array}$

by a limit procedure. The proof of Lemma 4.4 is completed. ∎

We can now easily prove our main result.

The proof of Theorem 1.2

A combination of Lemma 4.1 and Lemma 4.4 directly leads to our desired result.

Acknowledgement: This work is partially supported by the National Natural
Science Foundation of China (No. 11601215), Shandong Provincial Science
Foundation for Out- standing Youth (No. ZR2018JL005) and Project funded by
China Postdoctoral Science Foundation (No. 2019M650927, 2019T120168).

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