# Global dynamics in a chemotaxis model describing tumor angiogenesis
with/without mitosis in any dimensions

Jiawei Chu  School of Mathematics, South China University of Technology,
Guangzhou 510640, China<EMAIL_ADDRESS>,  Hai-Yang Jin  School
of Mathematics, South China University of Technology, Guangzhou 510640, China
<EMAIL_ADDRESS>and  Tian Xiang∗ Institute for Mathematical Sciences,
Renmin University of China, Bejing, 100872, China<EMAIL_ADDRESS>
###### Abstract.

In this work, we study the following Neumann-initial boundary value problem
for a three-component chemotaxis model describing tumor angiogenesis:

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+\xi_{1}\nabla\cdot(u\nabla w)+u(a-\mu u^{\theta}),&x\in\Omega,t>0,\\\
v_{t}=d\Delta v+\xi_{2}\nabla\cdot(v\nabla w)+u-v,&x\in\Omega,t>0,\\\ 0=\Delta
w+u-\bar{u},\ \ \int_{\Omega}w=0,\ \
\bar{u}:=\frac{1}{|\Omega|}\int_{\Omega}u,&x\in\Omega,t>0,\\\ \frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial
w}{\partial\nu}=0,&x\in\partial\Omega,t>0,\\\
u(x,0)=u_{0}(x),~{}v(x,0)=v_{0}(x),&x\in\Omega,\end{cases}$

in a bounded smooth but not necessarily convex domain
$\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with model parameters
$\xi_{1},\xi_{2},d,\theta>0,a,\chi,\mu\geq 0$. Based on subtle energy
estimates, we first identify two positive constants $\xi_{0}$ and $\mu_{0}$
such that the above problem allows only global classical solutions with
qualitative bounds provided one of the following conditions holds:

$(1)\ \xi_{1}\geq\xi_{0}\chi^{2};\ \ \ \ (2)\ \theta=1,\
\mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{0}\chi^{\frac{2}{5+n}};\ \ \ \ (3)\
\theta>1,\mu>0.$

Then, due to the obtained qualitative bounds, upon deriving higher order
gradient estimates, we show exponential convergence of bounded solutions to
the spatially homogeneous equilibrium (i) for $\mu$ large relative to
$\chi^{2}+\xi_{1}^{2}$ if $\mu>0$, (ii) for $d$ large if $a=\mu=0$ and (iii)
for merely $d>0$ if $\chi=a=\mu=0$. As a direct consequence of our findings,
all solutions to the above system with $\chi=a=\mu=0$ are globally bounded and
they converge to constant equilibrium, and therefore, no patterns can arise.

###### Key words and phrases:

Chemotaxis, tumor angiogenesis, convection, qualitative boundedness, global
stability.

###### 2000 Mathematics Subject Classification:

35A01, 35B40, 35B44, 35K57, 35Q92, 92C17

∗ Corresponding author.

##  1\. Introduction and statement of main results

To describe the branching of capillary sprouts during angiogenesis, Orme &
Chaplain [31] proposed the following reaction-advection-diffusion system

$\begin{cases}u_{t}=d_{1}\Delta u-\chi\nabla\cdot(u\nabla
v)+\xi_{1}\nabla\cdot(u\nabla w),\\\ v_{t}=d_{2}\Delta
v+\xi_{2}\nabla\cdot(v\nabla w)+\alpha u-\beta v,\\\ w_{t}=d_{3}\Delta
w+\gamma u-\delta w,\end{cases}$ (1.1)

with positive parameters
$d_{1},d_{2},d_{3},\chi,\xi_{1},\xi_{2},\beta,\delta,\alpha,\gamma$, where
$u,v$ and $w$ denote the density of endothelial cells (ECs), adhesive sites,
and the matrix (including fibronectin, laminin, and collagen IV),
respectively. Different from the classical mathematical models of tumor
angiogenesis with chemotaxis as the principle mechanism of cell motion ([8,
35]), the model (1.1) was proposed based on the experimental observations that
during angiogenesis process ECs secrete a matrix consisting of fibronectin,
laminin and collagen IV [33] and the movement of ECs is effected by the
distribution of adhesive sites on this matrix. More specifically, the
following two processes are essentially incorporated in (1.1) (see [31, 33]):

  * •

ECs secret matrix and adhesive sites;

  * •

The spreading of matrix with the convection of ECs and adhesive sites with it.

That is, the movement of ECs is governed by a combination of random motility,
chemotaxis and convection.

Before proceeding to our motivation and main results, we first recall some
most relevant results to the system (1.1) under homogeneous Neumann boundary
conditions (IBVP) and nonnegative initial conditions.

  * (i)

$\xi_{1}=\xi_{2}=0$: In this case, the first two components of (1.1) reduce to
the well-known classical (minimal) Keller-Segel chemotaxis model:

$\begin{cases}u_{t}=d_{1}\Delta u-\chi\nabla\cdot(u\nabla v),\\\
v_{t}=d_{2}\Delta v+\alpha u-\beta v,\\\ \end{cases}$ (1.2)

whose solution behaviors have been extensively studied in various perspectives
in the past five decades including boundedness, blow-up, large time behavior
and pattern formation. One can find more details from survey articles [10, 6,
5] and the references therein. More precisely, the boundedness and blowup of
solutions for (1.2) have been established in two or higher dimensions [30, 42,
40, 11] based on the following Lyapunov energy functional:

$\mathcal{E}_{1}(u,v)=d_{1}\int_{\Omega}u\ln
u-\chi\int_{\Omega}uv+\frac{\beta\chi}{2\alpha}\int_{\Omega}v^{2}+\frac{\chi
d_{2}}{2\alpha}\int_{\Omega}|\nabla v|^{2}.$

  * (ii)

$\xi_{2}=0$: This case means that the convection effect of matrix on the
adhesive sites is neglected, and then the system (1.1) reduces to the
following widely studied attraction-repulsion Keller-Segel (ARKS) model

$\begin{cases}u_{t}=d_{1}\Delta u-\chi\nabla\cdot(u\nabla
v)+\xi_{1}\nabla\cdot(u\nabla w),\\\ v_{t}=d_{2}\Delta v+\alpha u-\beta v,\\\
w_{t}=d_{3}\Delta w+\gamma u-\delta w.\end{cases}$ (1.3)

The ARKS model (1.3) has been proposed to describe the aggregation of
Microglia in Alzheimer’s disease in [27] and to describe quorum effect in
chemotaxis [32]. In one dimensional space, the existence of global boundedness
of classical solution [26, 14] and time-periodic patterns and steady states
patterns [24] were established. In high dimensional spaces ($n\geq 2$), it has
been found that the sign of $\Theta:=d_{2}\xi_{1}\gamma-d_{3}\chi\alpha$ plays
an important role in determining the solution behavior of (1.3). More
precisely, if $\Theta\geq 0$ (i.e., repulsion dominates or cancel attraction),
the 2D fully parabolic ARKS model [13, 25] or higher D parabolic-elliptic-
elliptic simplification of the ARKS model [39] admits only global bounded
solutions. However, if $\Theta<0$ (i.e., attraction prevails over repulsion),
based on the availability of Lyapunov functional, a critical mass phenomenon
has been found (see [20, 9, 15] for more details). Recently, in the repulsion
dominated case, i.e, $\Theta\geq 0$, the global stability of constant steady
state has been studied in [22, 17].

  * (iii)

$\xi_{1},\xi_{2}>0$: Due to the strong coupling of chemotaxis and convection
in a cascade-like manner, which increases the complexity of mathematical
analysis, the Lyapunov functional as constructed for the system (1.3) does not
work anymore. To the best of our knowledge, the existing results on the system
(1.1) seem at a rather rudimentary stage: in one dimensional space, the global
existence of classical solution was very recently established in [21] based on
semigroup estimate technique. Furthermore, based on an appropriate energy
functional, the 1D bounded solution was shown to converge to constant steady
state under some restrictions on the model parameters like $\xi_{2}$ is small
[18]. Very recently, for a parabolic-parabolic-elliptic simplified model in a
bounded convex domain $\Omega\subset\mathbb{R}^{n}(1\leq n\leq 3)$, Tao &
Winkler [38] used a Moser-type iteration to derive the global boundedness of
classical solution for large $\xi_{1}$ without qualitative information.

In summary of the above related results, some interesting questions naturally
arise:

  * (Q1)

It follows from [38] that large repulsive convection prevents any blow-up
phenomenon in $\leq 3$D (lower dimensional) convex domains. Hence, it is
nature to ask whether or not large repulsive convection can prevent blow-up of
solution in any dimensional bounded smooth but not necessarily convex domains.

  * (Q2)

In one dimensional space, due to nice Sobolev embeddings, a small
$\xi_{2}(\in(0,1])$ -independent upper bound of solution is available. This
makes the small $\xi_{2}$-global stability toward constant equilibrium
possible [18]. However, in higher dimensions, solution bounds typically depend
on $\xi_{2}$ (actually with a complex relation containing $\xi_{2}$ and its
inverse $\xi_{2}^{-1}$, cf. (1.7) and (1.8) for instance), and thus the method
used in [18] does not work anymore. Hence, it is challenging to study long
time dynamics of bounded solutions in higher dimensions.

  * (Q3)

Although there may be no significant increase in the rate of ECs mitosis
during the first stages of angiogenesis, the mitosis occurs after the first
spouts have formed [31]. Moreover, cell division also plays an essential role
when repairing and remodelling of large wounds [33]. Thus, it is interesting
and practically needed to study the effect of mitosis for the system (1.1).

To study the impact of convection more deeply and to provide relatively
complete answers for the three questions above, for simplicity and clarity,
based on the assumption that matrix diffuses much faster than adhesive sites
and endothelial cells, we shall use a quasi-stationary approximation procedure
as in [12, 41] ($\tilde{w}=w-\bar{w}$, and then the $w$-equation becomes
$d_{3}^{-1}\tilde{w}_{t}=\Delta\tilde{w}+\gamma d_{3}^{-1}(u-\bar{u})-\delta
d_{3}^{-1}\tilde{w}$; then assuming $\gamma$ has the same order as $d_{3}$ and
$\delta$ has lower order than $d_{3}$, and finally, sending
$d_{3}\rightarrow\infty$ and dropping the tilde notation) to lead to the
following version of parabolic-parabolic-elliptic problem:

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+\xi_{1}\nabla\cdot(u\nabla w)+u(a-\mu u^{\theta}),&x\in\Omega,t>0,\\\
v_{t}=d\Delta v+\xi_{2}\nabla\cdot(v\nabla w)+u-v,&x\in\Omega,t>0,\\\ 0=\Delta
w+u-\bar{u},\ \ \int_{\Omega}w=0,\ \
\bar{u}:=\frac{1}{|\Omega|}\int_{\Omega}u,&x\in\Omega,t>0,\\\ \frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial
w}{\partial\nu}=0,&x\in\partial\Omega,t>0,\\\
u(x,0)=u_{0}(x),~{}v(x,0)=v_{0}(x),&x\in\Omega,\end{cases}$ (1.4)

where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with smooth boundary.
Here we keep the parameters $\chi,\xi_{1},\xi_{2}$ as above and simplify other
parameters in an obvious way for convenience. The kinetic term $u(a-\mu
u^{\theta})$ with $a,\mu\geq 0,\theta>0$ is incorporated to see the effect of
ECs mitosis. We shall henceforth assume that

$(u_{0},v_{0})\in C^{0}(\bar{\Omega})\times
W^{1,\infty}~{}~{}\text{with}~{}~{}u_{0}\geq 0,v_{0}\geq
0,~{}\text{and}~{}u_{0}\not\equiv 0.$ (1.5)

Then our main findings on qualitative boundedness and convergence are stated
as follows.

######  Theorem 1.1 (Qualitative boundedness).

Let $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ be a bounded and smooth domain, the
model parameters $\xi_{1},\xi_{2},d,\theta>0,a,\chi,\mu\geq 0$, and, let the
initial data $(u_{0},v_{0})$ satisfy the regularity (1.5). Then there exist
positive constants $\xi_{0}$ and $\mu_{0}$ depending on
$d,\xi_{2},n,u_{0},v_{0},\Omega$ such that the IBVP (1.4) admits a unique
global classical solution provided one of the following conditions holds:

  * •

(1) $\xi_{1}\geq\xi_{0}\chi^{2}$; (2) $\theta=1,\ \mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{0}\chi^{\frac{2}{5+n}}$; (3)
$\theta>1,\mu>0$.

Moreover, the global solution is qualitatively bounded in the following way:

$\|u(\cdot,t)\|_{L^{\infty}}+\|v(\cdot,t)\|_{W^{1,\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\leq
M,\quad\quad\quad\forall t>0,$ (1.6)

where, up to a multiplier depending only on $n,u_{0},v_{0}$ and $\Omega$, the
positive constant $M$ is explicitly expressible in terms of the model
parameters $d,\chi_{1},\xi_{1},\xi_{2},\mu$, see Lemmas 3.1, 3.5, 3.6 and 3.7.
In particular, the qualitative bounds for $\|v\|_{L^{\infty}}$ and $\|\nabla
w\|_{L^{\infty}}$, crucial to derive large time behaviors of bounded solutions
for (1.4), are bounded as follows:

$\begin{split}\|v(\cdot,t)\|_{L^{\infty}}&\leq
K_{1}\begin{cases}\left(1+\frac{1}{\xi_{2}}\right)\left(1+\xi_{2}+(\frac{1}{d})^{\frac{n}{2}}\xi_{2}^{1+\frac{n}{2}}\right),&\text{
if }\mu=0,\\\\[7.11317pt]
\left(1+(\frac{1}{\mu})^{\frac{1}{\theta}}+\frac{1}{\xi_{2}}\right)\left(1+(\frac{1}{\mu})^{\frac{1}{\theta}}\xi_{2}+(\frac{1}{d})^{\frac{n}{2}}[(\frac{1}{\mu})^{\frac{1}{\theta}}\xi_{2}]^{1+\frac{n}{2}}\right),&\text{
if }\mu>0,\end{cases}\\\ &:=M_{0},\end{split}$ (1.7)

and

$\|\nabla w(\cdot,t)\|_{L^{\infty}}\leq
K_{2}\left(1+\left(1+d_{\Omega}M_{0}^{2(n+1)}\right)d^{-1}\chi^{2}M_{0}^{1-n}+M_{1}^{c}(n)\right)^{\frac{1}{n+1}},$
(1.8)

where $M_{1}^{c}(n)=M_{1}^{c}$ is defined by

$M_{1}^{c}(n)=\begin{cases}\xi_{1},\quad\quad\quad\quad\quad\quad\quad\text{if
}\mu=0,\ \xi_{1}\geq\xi_{0}\chi^{2},\\\\[7.11317pt]
M_{\mu}(1),\quad\quad\quad\quad\quad\text{if }\theta=1,\ \mu>\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{0}\chi^{\frac{2}{5+n}},\\\\[7.11317pt]
M_{\mu}(\theta)+\frac{(\theta-1)}{\mu^{\frac{n+2}{\theta-1}}}\left[(1+\frac{1}{d^{n+2}})(1+M_{0}\xi_{2})M_{0}\chi^{2}\right]^{\frac{n+1+\theta}{\theta-1}},\
\text{if }\theta>1,\mu>0,\end{cases}$ (1.9)

the function $M_{\mu}$ and the symbol $d_{\Omega}=d1_{\Omega}$ with
$1_{\Omega}$ being the indicator whether $\Omega$ is non-convex are defined by

$M_{\mu}(\theta)=\left(1+\xi_{1}(\frac{1}{\mu})^{\frac{1}{\theta}}+(\frac{1}{\mu})^{\frac{1}{\theta}}\right)(\frac{1}{\mu})^{\frac{n+1}{\theta}},\
\ d_{\Omega}=d1_{\Omega}=\begin{cases}0,&\text{ if }\Omega\text{ is
convex},\\\ d,&\text{ if }\Omega\text{ is non-convex}.\end{cases}$ (1.10)

######  Remark 1.2.

When $\Omega\subset\mathbb{R}^{1}$ is an open interval, using simpler
arguments than those of [18, 21], one can easily obtain global boundedness
without any parameter restrictions. For fixed parameters and initial data, we
also mention that the infimums of $\xi_{0}$ and $\mu_{0}$ depend indeed on
$\frac{n}{2}$ instead of $n$, see Lemma 3.6; this is comparable to the widely
known $L^{\frac{n}{2}+}$-criterion [6, 43]. Moreover, we note that the upper
bounds for $\|v\|_{L^{\infty}}$ and $\|\nabla w\|_{L^{\infty}}$, cf. (1.7) and
(1.8), are bounded for large $d$ and, in particular, it can be non-increasing
in $d$ in the case that $\Omega$ is convex. This makes the study of global
stability possible in the case of $a=\mu=0$.

Thanks in particular to the qualitative bounds for $\|v\|_{L^{\infty}}$ and
$\|\nabla w\|_{L^{\infty}}$ in (1.7) and (1.8), upon successfully deriving
higher order gradient estimates, we are able to prove convergence and
exponential convergence rate of bounded solutions in the following ways.

######  Theorem 1.3 (Global stability).

The global bounded classical solution $(u,v,w)$ obtained from Theorem 1.1
enjoys the following convergence properties:

  * (C1)

In the case of $a=\mu=0$, there exists $d_{0}(\chi)\geq 0$ with $d_{0}(0)=0$
depending on $n,u_{0},v_{0},\xi_{1},\xi_{2},\chi$ such that whenever $d\geq
d_{0}(\chi)$ with $\chi>0$, the solution $(u,v,w)$ converges uniformly to
$(\bar{u}_{0},\bar{u}_{0},0)$:

$\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}+\|v(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\to
0\ \ \mathrm{as}\ \ t\to\infty.$ (1.11)

If, in addition, $d>d_{0}(\chi)$, the above convergence is exponential: for
some $K_{3},\zeta>0$,

$\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}+\|v(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\leq
K_{3}e^{-\zeta t},\ \ \forall t>0.$ (1.12)

  * (C2)

In the case of $\mu>0$, let $C_{p}$ denote the Poincaré constant defined by

$C_{p}^{-2}=\inf\left\\{\int_{\Omega}|\nabla
w|^{2}:\int_{\Omega}w=0,\int_{\Omega}w^{2}=1\right\\}$ (1.13)

and let

$\Lambda(z)=\frac{d\chi^{2}+d^{2}C^{2}_{p}\xi_{1}^{2}+C^{2}_{p}\xi_{2}^{2}z}{2d^{2}a^{\frac{\theta-2}{\theta}}}.$
(1.14)

Then, whenever

$\mu>\max\left\\{\Lambda^{\frac{\theta}{2}}\left(\sup_{\mu>1}M_{0}^{2}\right),\
\Lambda^{\frac{\theta}{6+n}}\left(\sup_{0<\mu\leq
1}\left(M_{0}^{2}\mu^{\frac{4+n}{\theta}}\right)\right)\right\\},$ (1.15)

the solution $(u,v,w)$ converges exponentially to
$((\frac{a}{\mu})^{\frac{1}{\theta}},(\frac{a}{\mu})^{\frac{1}{\theta}},0)$:
for some $K_{4},\eta>0$,

$\|u(\cdot,t)-(\frac{a}{\mu})^{\frac{1}{\theta}}\|_{L^{\infty}}+\|v(\cdot,t)-(\frac{a}{\mu})^{\frac{1}{\theta}}\|_{L^{\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\leq
K_{4}e^{-\eta t},\ \ \ \forall t>0.$ (1.16)

In the above texts, we have used the following short notations like

$\|f(\cdot,t)\|_{L^{p}}=\left(\int_{\Omega}|f(x,t)|^{p}dx\right)^{\frac{1}{p}}=\left(\int_{\Omega}|f(\cdot,t)|^{p}\right)^{\frac{1}{p}}.$

######  Remark 1.4.

  * (i)

Compared to the global boundedness in [38], we first remove the technical
assumption that $\Omega$ is convex, and then we provide more qualitative
information by digging out the dependence of solution upper bounds on the
involving model parameters. Lastly, we extend lower dimensional spaces to any
dimensional ones.

  * (ii)

Even though we cannot apply the arguments in [18] to obtain large time
behaviors of bounded solutions in that fashion, due to especially the
qualitative bounds for $\|v\|_{L^{\infty}}$ and $\|\nabla w\|_{L^{\infty}}$ in
(1.7) and (1.8), upon successfully establishing higher order gradient
estimates, we are able to show convergence and exponential convergence of
bounded solutions in qualitative ways.

As a direct consequence of our main results, we obtain unconditional
boundedness and convergence to constant steady states of (1.4) with
$\chi=a=\mu=0$.

######  Corollary 1.5.

All solutions $(u,v,w)$ to the IBVP (1.4) with $\chi=0$ are globally bounded.
Moreover, if $a=\mu=0$, they converge exponentially to
$(\bar{u}_{0},\bar{u}_{0},0)$ as $t\rightarrow\infty$.

It can be easily seen from Theorem 1.1 that the dynamics of (1.4) with small
$\xi_{1}$ or simply $\xi_{1}=0$ remain largely open. We shall leave this as a
future project.

In the rest of this section, we outline the plan as well as main ideas of this
article.

In Section 2, we first state the local well-posedness and extensibility
criterion and then derive some basic properties for (1.4). Next, we extend the
interpolation type inequalities in [38, Lemma 4.2] to general setting in Lemma
2.4. Finally, we collect some abstract functionals inequalities including
well-known smoothing $L^{p}$-$L^{q}$ estimates of the Neumann heat group in
$\Omega$, etc.

In section 3, we aim to show the proof of qualitative boundedness in Theorem
1.1. Our subtle analysis, inspired from [38], begins with the qualitative
control of $\|v\|_{L^{\infty}}$ via Moser-iteration in a careful manner, cf.
Lemma 3.1. Then, thanks to the generalized interpolation type inequalities in
Lemma 2.4, under one of the conditions in Theorem 1.1 and upon some skillful
treatments, we successfully establish a key ODI (ordinary differential
inequality) for the time derivative of the coupled quantity
$\|u\|_{L^{k}}^{k}+\|\nabla v\|_{L^{2k}}^{2k}$ for some $k>n/2$, which allows
us to derive qualitative bounds for $\|u\|_{L^{k}}+\|\nabla v\|_{L^{2k}}$, cf.
Lemma 3.5. Thereafter, with subtle analysis via semigroup estimates, elliptic
estimates and Sobolev embeddings, we finally conclude the desired uniform-in-
time qualitative bounds as stated in (1.6).

In Section 4, we proceed further to derive Schauder type estimates of $u$ and
$L^{2n}$-estimate of $\nabla u$ so as to study long time dynamics of bounded
solutions to (1.4). Since both the $u$\- and $v$-equation in (1.4) have cross-
diffusions, the derivation of boundedness of $\|\nabla u\|_{L^{2n}}$ becomes
lengthy and technical. Roughly speaking, motivated from [23, Section 3.3], we
first establish an ODI for the time evolution of the coupled quantity
$\|u\|_{L^{2}}^{2}+\|\Delta v\|_{L^{2}}^{2}$ to obtain a bound for $\|\Delta
v\|_{L^{2}}$. And then, we directly derive an ODI for $\|\nabla
u\|_{L^{2}}^{2}$ so as to obtain a bound of $\|\nabla u\|_{L^{2}}$. This
enables us to handle the emerging boundary integral and thus allows us to
derive an ODI involving $\|\Delta v\|_{L^{2n}}^{2n}$ for the time derivative
of $\|\nabla u\|_{L^{2n}}^{2n}$. Finally, applying the widely known maximal
Sobolev regularity to the $v$-equation in (1.4), we obtain the boundedness of
$\|\nabla u\|_{L^{2n}}$, see Lemma 4.2.

With qualitative bounds in Section 3 and enhanced regularity properties in
Section 4, in Section 5, we aim to study the large time behavior of bounded
solutions to (1.4) . In the absence of ECs mitosis ($a=\mu=0$), based on the
important fact that upper bounds of $\|v\|_{L^{\infty}}$ and $\|\nabla
w\|_{L^{\infty}}$ are bounded for $d$ away from zero (cf. (1.7) and (1.8)),
for $d$ suitably large, we deduce a Lyapunov functional for the coupled
quantity

$\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2},$

which enables us to derive the $L^{2}$\- convergence and decay rate of
$(u-\bar{u}_{0},\nabla v)$ to $(0,0)$. Then using the Gagliardo-Nirenberg
inequality together with the enhanced regularity of $(u,v)$ and standard
elliptic estimates, we achieve the convergence properties of bounded solution
$(u,v,w)$ as in (1.11) and (1.12) of Theorem 1.3, see details in Section 5.2.

The convergence analysis for the case $a,\mu>0$ follows in a similar manner.
Indeed, for $\mu$ satisfying (1.15), we are able to derive a Lyapunov
functional for the coupled quantity

$\int_{\Omega}(u-b-b\ln\frac{u}{b})+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2},\
\ \ b=:(\frac{a}{\mu})^{\frac{1}{\theta}},$

which yields the key starting $L^{2}$-convergence of $(u-b,v-b)$ to $(0,0)$.
Then we can easily use GN interpolation inequality to lift this
$L^{2}$-convergence to $L^{\infty}$-convergence. Finally, the convergence
property of $w$ follows from the elliptic estimate applied to the $w$-equation
in (1.4), thus achieving (1.16) of Theorem 1.3, see Section 5.2 for details.

##  2\. Local existence and preliminaries

In the sequel, the integral $\int_{\Omega}f(x)dx$ and $\|f\|_{L^{p}(\Omega)}$
will be abbreviated as $\int_{\Omega}f$ and $\|f\|_{L^{p}}$, respectively. The
generic constants $c_{i}$ (defined within the proof of lemmas) or $C_{i}$
(defined in the statements of lemmas) for $i=1,2,\cdots$, depending on
$n,\Omega$ and the initial data $u_{0},v_{0}$ but they are independent of $t$
and the model parameters $\chi,\xi_{1},\xi_{2},d,\mu$, will vary line-by-line.
The existence and uniqueness of local solutions of (1.4) can be easily shown
in a fixed point theorem framework by means of the Amann’s theorems [3, 4] and
the parabolic/elliptic regularity theory, as similarly demonstrated in [18,
21, 38, 41].

######  Lemma 2.1 (Local existence).

Let $\Omega\subset\mathbb{R}^{n}(n\geq 1)$ be a bounded and smooth domain, the
model parameters $\chi,\xi_{1},\xi_{2},d>0,a,\mu,\theta\geq 0$, and, let the
initial data $(u_{0},v_{0})$ satisfy (1.5). Then there exist a maximal time
$T_{max}\in(0,\infty]$ and a unique triple $(u,v,w)$ of functions with $u$ and
$v$ positive which solves (1.4) classically on
$\bar{\Omega}\times(0,T_{max})$, and satisfies

$\begin{cases}u\in C^{0}(\bar{\Omega}\times[0,T_{max}))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max})),\\\
v\in\displaystyle\cap_{p>n}C^{0}([0,T_{max});W^{1,p}(\Omega))\cap
C^{2,1}(\bar{\Omega}\times(0,T_{max})),\\\ w\in
C^{2,0}(\bar{\Omega}\times(0,T_{max})).\end{cases}$

Moreover, if $T_{max}<\infty$, then, for any $p>\max\\{n,2\\}$,

$\lim\limits_{t\nearrow
T_{max}}\sup\left\\{\|u(\cdot,t)\|_{L^{\infty}}+\|v(\cdot,t)\|_{W^{1,p}}\right\\}=\infty.$
(2.1)

######  Lemma 2.2 (Young’s inequality with $\varepsilon$).

Let $1<p,q<\infty$, $\frac{1}{p}+\frac{1}{q}=1$. Then

$XY\leq\varepsilon X^{P}+\frac{Y^{q}}{q\left(\varepsilon
p\right)^{\frac{q}{p}}}\ \ (X,Y>0,\ \varepsilon>0).$

######  Lemma 2.3.

Let $(u,v,w)$ be a solution of (1.4) obtained in Lemma 2.1. Then

$|\Omega|\bar{u}=\|u(\cdot,t)\|_{L^{1}}\leq m_{1},\ \ \ \forall
t\in(0,T_{max}),$ (2.2)

where

$m_{1}:=\begin{cases}\|u_{0}\|_{L^{1}},&\mathrm{if}\ \ a=\mu=0,\\\\[7.11317pt]
\|u_{0}\|_{L^{1}}+(1+a)^{\frac{1+\theta}{\theta}}(\frac{1}{\mu})^{\frac{1}{\theta}}(\frac{2}{\theta+1})^{\frac{1}{\theta}}\frac{\theta}{\theta+1}|\Omega|,&\mathrm{if}\
\ \mu>0,\end{cases}$

and

$\|v(\cdot,t)\|_{L^{1}}\leq m_{1}+\|v_{0}\|_{L^{1}}.$ (2.3)

###### Proof.

Integrating the first and second equations of (1.4) over $\Omega$
respectively, one has

$\frac{d}{dt}\int_{\Omega}u+\mu\int_{\Omega}u^{\theta+1}=a\int_{\Omega}u,$
(2.4)

and

$\frac{d}{dt}\int_{\Omega}v+\int_{\Omega}v=\int_{\Omega}u.$ (2.5)

If $a=\mu=0$, one can easily check that (2.2) and (2.3) hold by integrating
(2.4) and (2.5).

Next, if $\mu>0$, it follows from the Young’s inequality with $\varepsilon$ in
Lemma 2.2 that

$\displaystyle(1+a)\int_{\Omega}u\leq\frac{\mu}{2}\int_{\Omega}u^{\theta+1}+(1+a)^{\frac{1+\theta}{\theta}}(\frac{1}{\mu})^{\frac{1}{\theta}}(\frac{2}{\theta+1})^{\frac{1}{\theta}}\frac{\theta}{\theta+1}|\Omega|,$

which, upon being substituted into (2.4), gives

$\frac{d}{dt}\int_{\Omega}u+\int_{\Omega}u+\frac{\mu}{2}\int_{\Omega}u^{\theta+1}\leq(1+a)^{\frac{1+\theta}{\theta}}(\frac{1}{\mu})^{\frac{1}{\theta}}(\frac{2}{\theta+1})^{\frac{1}{\theta}}\frac{\theta}{\theta+1}|\Omega|.$
(2.6)

Then multiplying (2.6) by the factor $e^{t}$ and then integrating, we simply
get

$\int_{\Omega}u\leq(1+a)^{\frac{1+\theta}{\theta}}(\frac{1}{\mu})^{\frac{1}{\theta}}(\frac{2}{\theta+1})^{\frac{1}{\theta}}\frac{\theta}{\theta+1}|\Omega|+\int_{\Omega}u_{0},$

which entails (2.2). Then substituting (2.2) into (2.5), one has

$\frac{d}{dt}\int_{\Omega}v+\int_{\Omega}v\leq(1+a)^{\frac{1+\theta}{\theta}}(\frac{1}{\mu})^{\frac{1}{\theta}}(\frac{2}{\theta+1})^{\frac{1}{\theta}}\frac{\theta}{\theta+1}|\Omega|+\int_{\Omega}u_{0}.$
(2.7)

Solving this ODI or using the Grönwall’s inequality again, then (2.7) implies
(2.3). ∎

For our later boundedness purpose, let us generalize the interpolation type
inequalities in [38, Lemma 4.2] to the general case as follows.

######  Lemma 2.4.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 1)$ be a bounded and smooth domain and
let $g,h\in C^{2}(\bar{\Omega})$ with $\frac{\partial
g}{\partial\nu}|_{\partial\Omega}=\frac{\partial
h}{\partial\nu}|_{\partial\Omega}=0$. Then, for all $p\geq 1$,

$\left|\int_{\Omega}|\nabla g|^{2p-2}\nabla g\cdot\nabla(\nabla g\cdot\nabla
h)\right|\leq\left(\frac{\sqrt{n}}{2p}+1\right)\|\nabla
g\|^{2p}_{L^{2(p+1)}}\|D^{2}h\|_{L^{p+1}},$ (2.8)

and

$\begin{split}&\left|\int_{\Omega}g\Delta h\nabla\cdot(|\nabla g|^{2p-2}\nabla
g)\right|\\\ &\leq\left(2(p-1)+\sqrt{n}\right)\|g\|_{L^{\infty}}\|\nabla
g\|^{p-1}_{L^{2(p+1)}}\|\Delta h\|_{L^{p+1}}\left(\int_{\Omega}|\nabla
g|^{2p-2}|D^{2}g|^{2}\right)^{\frac{1}{2}},\end{split}$ (2.9)

as well as

$\int_{\Omega}|\nabla
g|^{2(p+1)}\leq\left(2p+\sqrt{n}\right)^{2}\|g\|^{2}_{L^{\infty}}\int_{\Omega}|\nabla
g|^{2(p-1)}|D^{2}g|^{2}.$ (2.10)

###### Proof.

Using the following two facts

$\nabla(\nabla g\cdot\nabla h)=D^{2}g\cdot\nabla h+D^{2}h\cdot\nabla g,$

and

$\nabla|\nabla g|^{2p}=p|\nabla g|^{2(p-1)}\nabla|\nabla g|^{2}=2p|\nabla
g|^{2(p-1)}D^{2}g\cdot\nabla g,$ (2.11)

we thus use the symmetry of $D^{2}g$ and integration by parts formula to
derive that

$\begin{split}&\int_{\Omega}|\nabla g|^{2p-2}\nabla g\cdot\nabla(\nabla
g\cdot\nabla h)\\\ &=\int_{\Omega}|\nabla g|^{2p-2}\nabla
g\cdot\left(D^{2}g\cdot\nabla h+D^{2}h\cdot\nabla g\right)\\\
&=\int_{\Omega}|\nabla g|^{2p-2}(D^{2}g\cdot\nabla g)\cdot\nabla
h+\int_{\Omega}|\nabla g|^{2p-2}\nabla g\cdot(D^{2}h\cdot\nabla g)\\\
&=\frac{1}{2p}\int_{\Omega}\nabla|\nabla g|^{2p}\cdot\nabla
h+\int_{\Omega}|\nabla g|^{2p-2}\nabla g\cdot(D^{2}h\cdot\nabla g)\\\
&=-\frac{1}{2p}\int_{\Omega}|\nabla g|^{2p}\cdot\Delta h+\int_{\Omega}|\nabla
g|^{2p-2}\nabla g\cdot(D^{2}h\cdot\nabla g).\end{split}$ (2.12)

Noting the fact $|\Delta h|\leq\sqrt{n}|D^{2}h|$ and using Hölder’s
inequality, from (2.12) we have

$\displaystyle\left|\int_{\Omega}|\nabla g|^{2p-2}\nabla g\cdot\nabla(\nabla
g\cdot\nabla h)\right|\leq$
$\displaystyle\frac{\sqrt{n}}{2p}\int_{\Omega}|\nabla
g|^{2p}|D^{2}h|+\int_{\Omega}|\nabla g|^{2p}|D^{2}h|$ $\displaystyle\leq$
$\displaystyle(\frac{\sqrt{n}}{2p}+1)\int_{\Omega}|\nabla g|^{2p}|D^{2}h|$
$\displaystyle\leq$ $\displaystyle(\frac{\sqrt{n}}{2p}+1)\|\nabla
g\|^{2p}_{L^{2(p+1)}}\|D^{2}h\|_{L^{p+1}},$

which is our desired estimate (2.8). Similarly, we infer that

$\displaystyle\left|\int_{\Omega}g\Delta h\nabla\cdot(|\nabla g|^{2p-2}\nabla
g)\right|$ $\displaystyle=$ $\displaystyle\left|\int_{\Omega}g\Delta
h\nabla(|\nabla g|^{2p-2})\cdot\nabla g+\int_{\Omega}g\Delta h|\nabla
g|^{2p-2}\Delta g\right|$ $\displaystyle=$
$\displaystyle\left|2(p-1)\int_{\Omega}g\Delta h|\nabla g|^{2(p-2)}\nabla
g\cdot(D^{2}g\cdot\nabla g)+\int_{\Omega}g\Delta h|\nabla g|^{2p-2}\Delta
g\right|$ $\displaystyle\leq$
$\displaystyle(2(p-1)+\sqrt{n})\int_{\Omega}|g|\cdot|\Delta h|\cdot|\nabla
g|^{2p-2}\cdot|D^{2}g|$ $\displaystyle\leq$
$\displaystyle(2(p-1)+\sqrt{n})\|g\|_{L^{\infty}}\|\Delta
h\|_{L^{p+1}}\|\nabla g\|^{p-1}_{L^{2(p+1)}}\||\nabla
g|^{p-1}|D^{2}g|\|_{L^{2}},$

which gives rise to (2.9).

Finally, we use (2.11) to estimate the term $\int_{\Omega}|\nabla g|^{2(p+1)}$
as follows:

$\displaystyle\int_{\Omega}|\nabla g|^{2(p+1)}$
$\displaystyle=\int_{\Omega}|\nabla g|^{2p}\nabla g\cdot\nabla g$
$\displaystyle=-\int_{\Omega}g|\nabla g|^{2p}\Delta
g-\int_{\Omega}g\nabla(|\nabla g|^{2p})\cdot\nabla g$
$\displaystyle=-\int_{\Omega}g|\nabla g|^{2p}\Delta g-2p\int_{\Omega}g|\nabla
g|^{2(p-1)}\nabla g\cdot(D^{2}g\cdot\nabla g)$
$\displaystyle\leq(2p+\sqrt{n})\int_{\Omega}|g|\cdot|\nabla
g|^{2p}\cdot|D^{2}g|$
$\displaystyle\leq(2p+\sqrt{n})\|g\|_{L^{\infty}}\left(\int_{\Omega}|\nabla
g|^{2(p+1)}\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla
g|^{2(p-1)}|D^{2}g|^{2}\right)^{\frac{1}{2}},$

which upon simple algebraic manipulations entails (2.10). ∎

Now, for convenience of reference, we collect the well-known smoothing
$L^{p}$-$L^{q}$ estimates of the Neumann heat group in $\Omega$, which can be
found in [7, 40].

######  Lemma 2.5.

Let $(e^{t\Delta})_{t\geq 0}$ be the Neumann heat semigroup in $\Omega$, and
let $\lambda_{1}>0$ denote the first nonzero eigenvalue of $-\Delta$ in
$\Omega$ under Neumann boundary conditions. Then there exist some positive
constants $c_{i}~{}(i=1,2,3)$ depending only on $\Omega$ such that:

  * (i)

If $1\leq q\leq p\leq\infty$, then

$\|e^{t\Delta}f\|_{L^{p}}\leq
c_{1}\left(1+t^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|f\|_{L^{q}}~{}\text{for~{}all}~{}t>0$
(2.13)

holds for all $f\in L^{q}(\Omega)$ satisfying $\int_{\Omega}f=0$.

  * (ii)

If $1\leq q\leq p\leq\infty$, then

$\|\nabla e^{t\Delta}f\|_{L^{p}}\leq
c_{2}\left(1+t^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|f\|_{L^{q}}~{}\text{for~{}all}~{}t>0$

is valid for all $f\in L^{q}(\Omega)$.

  * (iii)

If $2\leq q\leq p<\infty$, then

$\|\nabla e^{t\Delta}f\|_{L^{p}}\leq
c_{3}\left(1+t^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|\nabla
f\|_{L^{q}}~{}\text{for~{}all}~{}t>0$

is true for all $f\in W^{1,p}(\Omega)$.

  * (iv)

If $1<q\leq p\leq\infty$, then

$\|e^{t\Delta}\nabla\cdot f\|_{L^{p}}\leq
c_{4}\left(1+t^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|f\|_{L^{q}}~{}\text{for~{}all}~{}t>0$
(2.14)

is valid for all $f\in(W^{1,p}(\Omega))^{n}$.

######  Lemma 2.6 ([17]).

Let $f(x,t)$ be a positive function for $(x,t)\in\Omega\times(0,\infty)$ and
define $\bar{f}=\frac{1}{|\Omega|}\int_{\Omega}f$. Then it holds that

$0\leq\frac{1}{2\bar{f}}\|f-\bar{f}\|_{L^{1}}^{2}\leq\int_{\Omega}f\ln\frac{f}{\bar{f}}\leq\frac{1}{\bar{f}}\|f-\bar{f}\|_{L^{2}}^{2}.$

##  3\. Qualitative boundedness: Proof of Theorem 1.1

In this section, we are devoted to proving the qualitative boundedness in
terms of model parameters as stated in Theorem 1.1. To this end, we first use
the convection effect to show the boundedness of $\|v\|_{L^{\infty}}$ based on
some ideas in [38].

######  Lemma 3.1.

Let $(u,v,w)$ be the solution of (1.4) obtained from Lemma 2.1. Then

$\begin{split}&\|v(\cdot,t)\|_{L^{\infty}}\\\ &\leq
C_{0}\max\left\\{m_{1}+\|v_{0}\|_{L^{1}},\frac{1}{\xi_{2}},\|v_{0}\|_{L^{\infty}}\right\\}\left(1+m_{1}\xi_{2}+(\frac{1}{d})^{\frac{n}{2}}(m_{1}\xi_{2})^{1+\frac{n}{2}}\right):=M_{0},\end{split}$
(3.1)

for all $t\in[0,T_{\text{max}})$; here $C_{0}>0$ is defined in (3.13), which
depends only on $n$ and $\Omega$.

###### Proof.

For $p>1$, multiplying the second equation of (1.4) by $v^{p-1}$, integrating
the result over $\Omega$ by parts and combining the equation $\Delta
w=-u+\bar{u}$, we obtain that

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}v^{p}$
$\displaystyle=d\int_{\Omega}\Delta v\cdot
v^{p-1}+\xi_{2}\int_{\Omega}\nabla\cdot(v\nabla
w)v^{p-1}+\int_{\Omega}uv^{p-1}-\int_{\Omega}v^{p}$
$\displaystyle=-(p-1)d\int_{\Omega}v^{p-2}|\nabla
v|^{2}-\xi_{2}(p-1)\int_{\Omega}v^{p-1}\nabla v\cdot\nabla
w+\int_{\Omega}uv^{p-1}-\int_{\Omega}v^{p}$
$\displaystyle=-(p-1)d\int_{\Omega}v^{p-2}|\nabla
v|^{2}+\frac{\xi_{2}(p-1)}{p}\int_{\Omega}v^{p}\Delta
w+\int_{\Omega}uv^{p-1}-\int_{\Omega}v^{p}$
$\displaystyle=-(p-1)d\int_{\Omega}v^{p-2}|\nabla
v|^{2}+\frac{\xi_{2}(p-1)}{p}\int_{\Omega}v^{p}(-u+\bar{u})+\int_{\Omega}uv^{p-1}-\int_{\Omega}v^{p},$

which upon the fact $v^{p-2}|\nabla v|^{2}=\frac{4}{p^{2}}|\nabla
v^{\frac{p}{2}}|^{2}$ gives

$\begin{split}&\frac{d}{dt}\int_{\Omega}v^{p}+\frac{4d(p-1)}{p}\int_{\Omega}|\nabla
v^{\frac{p}{2}}|^{2}+\xi_{2}(p-1)\int_{\Omega}uv^{p}+p\int_{\Omega}v^{p}\\\
&=\xi_{2}(p-1)\bar{u}\int_{\Omega}v^{p}+p\int_{\Omega}uv^{p-1}.\end{split}$
(3.2)

Applying the Young’s inequality with $\varepsilon$ (cf. Lemma 2.2) and the
facts $\|u(\cdot,t)\|_{L^{1}}\leq m_{1}$ and
$\bar{u}=\frac{1}{|\Omega|}\int_{\Omega}u\leq\frac{m_{1}}{|\Omega|}$, we infer

$\begin{split}p\int_{\Omega}uv^{p-1}&\leq\xi_{2}(p-1)\int_{\Omega}uv^{p}+\xi_{2}^{-(p-1)}\int_{\Omega}u\\\
&\leq\xi_{2}(p-1)\int_{\Omega}uv^{p}+m_{1}\xi_{2}^{1-p},\end{split}$ (3.3)

and

$\xi_{2}(p-1)\bar{u}\int_{\Omega}v^{p}\leq\frac{\xi_{2}m_{1}(p-1)}{|\Omega|}\int_{\Omega}v^{p}.$
(3.4)

A substitution of (3.3) and (3.4) into (3.2) shows that

$\begin{split}&\frac{d}{dt}\int_{\Omega}v^{p}+\frac{4d(p-1)}{p}\int_{\Omega}|\nabla
v^{\frac{p}{2}}|^{2}+p\int_{\Omega}v^{p}\leq\frac{\xi_{2}m_{1}(p-1)}{|\Omega|}\int_{\Omega}v^{p}+m_{1}\xi_{2}^{1-p}.\end{split}$
(3.5)

For $\varepsilon>0$, there exists $c_{1}>0$ depending only on $n$ and $\Omega$
such that (cf. [39, 43])

$\|U\|_{L^{2}}^{2}\leq\varepsilon\|\nabla
U\|_{L^{2}}^{2}+c_{1}(1+\varepsilon^{-\frac{n}{2}})\|U\|_{L^{1}}^{2}.$ (3.6)

We choose $U=v^{\frac{p}{2}}$ and
$\varepsilon=\frac{4d|\Omega|}{p\xi_{2}m_{1}}$ in (3.6) to derive that

$\begin{split}&\frac{\xi_{2}m_{1}(p-1)}{|\Omega|}\int_{\Omega}v^{p}\\\
&\leq\frac{4(p-1)d}{p}\int_{\Omega}|\nabla
v^{\frac{p}{2}}|^{2}+\frac{\xi_{2}m_{1}c_{1}(p-1)}{|\Omega|}\left(1+\left(\frac{\xi_{2}m_{1}p}{4d|\Omega|}\right)^{\frac{n}{2}}\right)\left(\int_{\Omega}v^{\frac{p}{2}}\right)^{2}\\\
&\leq\frac{4(p-1)d}{p}\int_{\Omega}|\nabla
v^{\frac{p}{2}}|^{2}+c_{2}p(1+p^{\frac{n}{2}})\left(\int_{\Omega}v^{\frac{p}{2}}\right)^{2},\end{split}$
(3.7)

where
$c_{2}:=\frac{\xi_{2}m_{1}c_{1}}{|\Omega|}\max\\{1,\left(\frac{\xi_{2}m_{1}}{4d|\Omega|}\right)^{\frac{n}{2}}\\}$.
Then substituting (3.7) into (3.5) and noting the fact
$(1+p^{\frac{n}{2}})\leq 2(1+p)^{\frac{n}{2}}$, we obtain

$\begin{split}&\frac{d}{dt}\int_{\Omega}v^{p}+p\int_{\Omega}v^{p}\leq
2c_{2}p(1+p)^{\frac{n}{2}}\left(\int_{\Omega}v^{\frac{p}{2}}\right)^{2}+m_{1}\xi_{2}^{1-p},\end{split}$

which implies that

$\begin{split}\frac{d}{dt}\left(e^{pt}\int_{\Omega}v^{p}\right)\leq
2e^{pt}c_{2}p(1+p)^{\frac{n}{2}}\left(\int_{\Omega}v^{\frac{p}{2}}\right)^{2}+e^{pt}m_{1}\xi_{2}^{1-p}.\end{split}$
(3.8)

For any $T\in(0,T_{\text{max}})$, we integrate (3.8) over $[0,t]$ for $0\leq
t\leq T$ to obtain

$\begin{split}\int_{\Omega}v^{p}(x,t)&\leq\int_{\Omega}v_{0}^{p}+2c_{2}(1+p)^{\frac{n}{2}}\sup_{0\leq
t\leq
T}\left(\int_{\Omega}v^{\frac{p}{2}}(x,t)\right)^{2}+\frac{m_{1}\xi_{2}^{1-p}}{p},\end{split}$

which immediately yields

$\displaystyle\ \ \ \ \left(\int_{\Omega}v^{p}(x,t)\right)^{\frac{1}{p}}$
$\displaystyle\leq\left[2c_{2}(1+p)^{\frac{n}{2}}\sup_{0\leq t\leq
T}\left(\int_{\Omega}v^{\frac{p}{2}}(x,t)\right)^{2}+m_{1}\xi_{2}^{1-p}+|\Omega|\|v_{0}\|_{L^{\infty}}^{p}\right]^{\frac{1}{p}}$
$\displaystyle\leq\left(\max\left\\{2c_{2},m_{1}\xi_{2},|\Omega|\right\\}\right)^{\frac{1}{p}}(1+p)^{\frac{n}{2p}}\left\\{\sup_{0\leq
t\leq
T}\left(\int_{\Omega}v^{\frac{p}{2}}(x,t)\right)^{\frac{2}{p}}+\frac{1}{\xi_{2}}+\|v_{0}\|_{L^{\infty}}\right\\}.$

Therefore, it follows with $c_{3}:=3\max\\{2c_{2},m_{1}\xi_{2},|\Omega|,1\\}$
that

$\begin{split}&\ \ \ \ \max\left\\{\sup_{0\leq t\leq
T}\left(\int_{\Omega}v^{p}(x,t)\right)^{\frac{1}{p}},\ \ \frac{1}{\xi_{2}},\ \
\|v_{0}\|_{L^{\infty}}\right\\}\\\ &\leq
c_{3}^{\frac{1}{p}}(1+p)^{\frac{n}{2p}}\max\left\\{\sup_{0\leq t\leq
T}\left(\int_{\Omega}v^{\frac{p}{2}}(x,t)\right)^{\frac{2}{p}},\ \
\frac{1}{\xi_{2}},\ \ \|v_{0}\|_{L^{\infty}}\right\\}.\end{split}$ (3.9)

Upon setting

$H(p):=\max\left\\{\sup_{0\leq t\leq
T}\left(\int_{\Omega}v^{p}(x,t)\right)^{\frac{1}{p}},\ \ \frac{1}{\xi_{2}},\ \
\|v_{0}\|_{L^{\infty}}\right\\},$

then (3.9) becomes

$H(p)\leq c_{3}^{\frac{1}{p}}(1+p)^{\frac{n}{2p}}H(\frac{p}{2}),\ \ \ \forall
p\geq 2.$

Taking $p=2^{j}~{}(j=1,2,\cdots)$, we obtain inductively that

$\begin{split}H(2^{j})&\leq c_{3}^{2^{-j}}(1+2^{j})^{\frac{n}{2}\cdot
2^{-j}}H(2^{j-1})\\\ &\leq
c_{3}^{2^{-j}+2^{-(j-1)}}(1+2^{j})^{\frac{n}{2}\cdot
2^{-j}}(1+2^{j-1})^{\frac{n}{2}\cdot 2^{-(j-1)}}H(2^{j-2})\\\ &\leq
c_{3}^{\sum\limits_{i=0}^{2}2^{-(j-i)}}\cdot\prod_{k=j-2}^{j}(1+2^{k})^{\frac{1}{2^{k}}\cdot\frac{n}{2}}H(2^{j-3})\\\
&\cdots\cdots\\\ &\leq
c_{3}^{\sum\limits_{i=0}^{j-1}2^{-(j-i)}}\cdot\prod_{k=1}^{j}(1+2^{k})^{\frac{1}{2^{k}}\cdot\frac{n}{2}}H(1)\\\
&=c_{3}^{1-\frac{1}{2^{j}}}\prod_{k=1}^{j}(1+2^{k})^{\frac{1}{2^{k}}\cdot\frac{n}{2}}H(1).\end{split}$
(3.10)

On the one hand, using the fact that $\ln(1+z)\leq\sqrt{z}$ for all $z\geq 0$,
we have

$\ln\prod_{k=1}^{j}(1+2^{k})^{\frac{1}{2^{k}}}=\sum_{k=1}^{j}\frac{\ln(1+2^{k})}{2^{k}}\leq\sum_{k=1}^{\infty}\left(\frac{1}{\sqrt{2}}\right)^{k}\leq
6,\ \forall j=1,2,\cdots,$

and so,

$\lim_{j\to\infty}\prod_{k=1}^{j}(1+2^{k})^{\frac{1}{2^{k}}\cdot\frac{n}{2}}\leq
e^{3n}.$ (3.11)

Then the combination of (3.10), (3.11) and (2.3) with
$c_{2}:=\frac{\xi_{2}m_{1}c_{1}}{|\Omega|}\max\\{1,(\frac{\xi_{2}m_{1}}{4d|\Omega|})^{\frac{n}{2}}\\}$
gives

$\begin{split}\|v(\cdot,t)\|_{L^{\infty}}&\leq\lim_{j\to\infty}H(2^{j})\\\
&\leq c_{3}e^{3n}H(1)\\\
&=3\max\\{2c_{2},m_{1}\xi_{2},|\Omega|,1\\}e^{3n}\max\left\\{\sup_{0\leq t\leq
T}\int_{\Omega}v(\cdot,t),\ \frac{1}{\xi_{2}},\
\|v_{0}\|_{L^{\infty}}\right\\}\\\ &\leq
3e^{3n}\left(1+|\Omega|+m_{1}\xi_{2}+2c_{2}\right)\max\left\\{m_{1}+\
\|v_{0}\|_{L^{1}},\frac{1}{\xi_{2}},\ \|v_{0}\|_{L^{\infty}}\right\\}\\\ &\leq
C_{0}\max\left\\{m_{1}+\|v_{0}\|_{L^{1}},\frac{1}{\xi_{2}},\|v_{0}\|_{L^{\infty}}\right\\}\left((1+m_{1}\xi_{2}+(\frac{1}{d})^{\frac{n}{2}}(m_{1}\xi_{2})^{1+\frac{n}{2}}\right)\end{split}$
(3.12)

for all $t\in[0,T]$, where

$C_{0}:=3e^{3n}\max\left\\{1+|\Omega|,1+\frac{2c_{1}}{|\Omega|},\frac{2c_{1}}{(4|\Omega|)^{\frac{n}{2}}|\Omega|}\right\\}.$
(3.13)

Since $T\in(0,T_{\text{max}})$ is arbitrary and the upper bound in (3.12) is
independent of $T$, the desired estimate (3.1) follows from (3.12) and (3.13).
∎

###  3.1. Qualitative $L^{p}$-estimates

In this subsection, by means of the key estimate on $\|v\|_{L^{\infty}}$ in
(3.12), we shall establish further coupled $L^{p}$-energy estimates with
dependence on key parameters. Here, we shall stress that our arguments are
applicable to both cases: $a,\mu>0$ and $a=\mu=0$.

######  Lemma 3.2.

For $p>1$, the local-in-time solution $(u,v,w)$ of (1.4) obtained in Lemma 2.1
satisfies

$\begin{split}&\frac{d}{dt}\int_{\Omega}u^{p}+\frac{p(p-1)}{2}\int_{\Omega}u^{p-2}|\nabla
u|^{2}+\frac{\xi_{1}(p-1)}{2}\int_{\Omega}u^{p+1}+\frac{p\mu}{2}\int_{\Omega}u^{\theta+p}\\\
&\leq\frac{\chi^{2}p(p-1)}{2}\int_{\Omega}u^{p}|\nabla v|^{2}+M_{1}(p),\ \
\mathrm{for}\ \ t\in(0,T_{max}),\end{split}$ (3.14)

where

$M_{1}(p):=\begin{cases}\xi_{1}m_{1}^{p+1}\frac{(p-1)}{p+1}\cdot\left(\frac{2p}{|\Omega|(p+1)}\right)^{p}+\left(\frac{1}{\mu}\right)^{\frac{p}{\theta}}\left(\frac{2ap}{p+\theta}\right)^{\frac{p}{\theta}}\frac{ap\theta}{p+\theta}|\Omega|,&{\color[rgb]{0,0,0}a,\mu>0,}\\\\[7.11317pt]
\xi_{1}m_{1}^{p+1}\frac{(p-1)}{p+1}\cdot\left(\frac{2p}{|\Omega|(p+1)}\right)^{p},&a=\mu=0.\end{cases}$
(3.15)

###### Proof.

For $p>1$, multiplying the first equation of (1.4) by $u^{p-1}$ and
integrating the result over $\Omega$ by parts, we conclude that

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}u^{p}+(p-1)\int_{\Omega}u^{p-2}|\nabla
u|^{2}+\mu\int_{\Omega}u^{\theta+p}$
$\displaystyle=\chi(p-1)\int_{\Omega}u^{p-1}\nabla u\cdot\nabla
v-\xi_{1}(p-1)\int_{\Omega}u^{p-1}\nabla u\cdot\nabla w+a\int_{\Omega}u^{p}$
$\displaystyle=\chi(p-1)\int_{\Omega}u^{p-1}\nabla u\cdot\nabla
v+\frac{\xi_{1}(p-1)}{p}\int_{\Omega}u^{p}\Delta w+a\int_{\Omega}u^{p},$

which together with the fact $\Delta w=\bar{u}-u$ gives

$\begin{split}&\frac{1}{p}\frac{d}{dt}\int_{\Omega}u^{p}+(p-1)\int_{\Omega}u^{p-2}|\nabla
u|^{2}+\mu\int_{\Omega}u^{\theta+p}+\frac{\xi_{1}(p-1)}{p}\int_{\Omega}u^{p+1}\\\
=~{}&\chi(p-1)\int_{\Omega}u^{p-1}\nabla u\cdot\nabla
v+\frac{\xi_{1}(p-1)}{p}\bar{u}\int_{\Omega}u^{p}+a\int_{\Omega}u^{p}.\end{split}$
(3.16)

Applications of Hölder’s inequality and Young’s inequality yield that

$\begin{split}\chi(p-1)\int_{\Omega}u^{p-1}\nabla u\cdot\nabla
v&\leq\frac{(p-1)}{2}\int_{\Omega}u^{p-2}|\nabla
u|^{2}+\frac{\chi^{2}(p-1)}{2}\int_{\Omega}u^{p}|\nabla v|^{2},\end{split}$
(3.17)

and noting from
$\bar{u}=\frac{1}{|\Omega|}\int_{\Omega}u\leq\frac{m_{1}}{|\Omega|}$ that

$\begin{split}\frac{\xi_{1}(p-1)}{p}\bar{u}\int_{\Omega}u^{p}&\leq\frac{\xi_{1}(p-1)m_{1}}{p|\Omega|}\int_{\Omega}u^{p}\\\
&\leq\frac{\xi_{1}(p-1)}{2p}\int_{\Omega}u^{p+1}+{\color[rgb]{0,0,0}\xi_{1}m_{1}^{p+1}\frac{(p-1)}{p(p+1)}\left(\frac{2p}{|\Omega|(p+1)}\right)^{p},}\end{split}$
(3.18)

as well as, in the case of $a,\mu>0$,

$\begin{split}a\int_{\Omega}u^{p}&\leq\frac{\mu}{2}\int_{\Omega}u^{p+\theta}+\left(\frac{2ap}{\mu(p+\theta)}\right)^{\frac{p}{\theta}}\frac{{\color[rgb]{0,0,0}a}\theta}{p+\theta}|\Omega|.\\\
\end{split}$ (3.19)

Substituting (3.17), (3.18) and (3.19) into (3.16) and noting the definition
of $M_{1}(p)$ in (3.15), we end up with (3.14) directly, thus proving this
lemma. ∎

######  Lemma 3.3.

For $k\geq 1$, the local-in-time classical solution of (1.4) obtained in Lemma
2.1 satisfies, for $t\in[0,T_{\text{max}})$, that

$\begin{split}&\frac{d}{dt}\int_{\Omega}|\nabla v|^{2k}+2k\int_{\Omega}|\nabla
v|^{2k}+2kd\int_{\Omega}|\nabla v|^{2k-2}|D^{2}v|^{2}\\\ &\ \ \
+k(k-1)d\int_{\Omega}|\nabla v|^{2k-4}|\nabla|\nabla v|^{2}|^{2}\\\ &\leq
2k\xi_{2}\int_{\Omega}|\nabla v|^{2k-2}\nabla v\cdot\nabla(\nabla v\cdot\nabla
w)-2k\xi_{2}\int_{\Omega}v\Delta w\nabla\cdot(|\nabla v|^{2k-2}\nabla v)\\\ &\
\ \ \ +2k(2(k-1)+\sqrt{n})\int_{\Omega}u|\nabla
v|^{2k-2}|D^{2}v|+dk\int_{\partial\Omega}|\nabla
v|^{2k-2}\frac{\partial|\nabla v|^{2}}{\partial\nu}.\end{split}$ (3.20)

###### Proof.

For $k\geq 1$, differentiating the second equation of (1.4) and multiplying
the result by $|\nabla v|^{2k-2}\nabla v$, we obtain via integration by parts
that

$\displaystyle\frac{1}{2k}\frac{d}{dt}\int_{\Omega}|\nabla v|^{2k}$
$\displaystyle=\int_{\Omega}|\nabla v|^{2k-2}\nabla v\cdot\nabla(d\Delta
v+\xi_{2}\nabla\cdot(v\nabla w)-v+u)$ $\displaystyle=d\int_{\Omega}|\nabla
v|^{2k-2}\nabla v\cdot\nabla\Delta v+\xi_{2}\int_{\Omega}|\nabla
v|^{2k-2}\nabla v\cdot\nabla(\nabla\cdot(v\nabla w))$ (3.21)
$\displaystyle\;\;\;+\int_{\Omega}|\nabla v|^{2k-2}\nabla v\cdot\nabla
u-\int_{\Omega}|\nabla v|^{2k}$
$\displaystyle=:I_{1}+I_{2}+I_{3}-\int_{\Omega}|\nabla v|^{2k}.$

Using the identity $\nabla v\cdot\nabla\Delta v=\frac{1}{2}\Delta|\nabla
v|^{2}-|D^{2}v|^{2}$ and integrating by parts, we compute

$\begin{split}I_{1}&=\frac{d}{2}\int_{\Omega}|\nabla v|^{2k-2}\Delta|\nabla
v|^{2}-d\int_{\Omega}|\nabla v|^{2k-2}|D^{2}v|^{2}\\\
&=\frac{d}{2}\int_{\partial\Omega}|\nabla v|^{2k-2}\frac{\partial|\nabla
v|^{2}}{\partial\nu}-\frac{(k-1)d}{2}\int_{\Omega}|\nabla
v|^{2k-4}|\nabla|\nabla v|^{2}|^{2}\\\ &\ \ \ -d\int_{\Omega}|\nabla
v|^{2k-2}|D^{2}v|^{2}.\end{split}$ (3.22)

Similarly, we use integration by parts to rewrite $I_{2}$ as follows:

$\displaystyle I_{2}$ $\displaystyle=\xi_{2}\int_{\Omega}|\nabla
v|^{2k-2}\nabla v\cdot\nabla(\nabla v\cdot\nabla w+v\Delta w)$ (3.23)
$\displaystyle=\xi_{2}\int_{\Omega}|\nabla v|^{2k-2}\nabla v\cdot\nabla(\nabla
v\cdot\nabla w)+\xi_{2}\int_{\Omega}|\nabla v|^{2k-2}\nabla
v\cdot\nabla(v\Delta w)$ $\displaystyle=\xi_{2}\int_{\Omega}|\nabla
v|^{2k-2}\nabla v\cdot\nabla(\nabla v\cdot\nabla
w)-\xi_{2}\int_{\Omega}v\Delta w\nabla\cdot(|\nabla v|^{2k-2}\nabla v).$

As for $I_{3}$, using the fact $|\Delta v|\leq\sqrt{n}|D^{2}v|$ and the
identity (2.11), we have

$\displaystyle I_{3}$ $\displaystyle=\int_{\Omega}|\nabla v|^{2k-2}\nabla
v\cdot\nabla u$ $\displaystyle=-\int_{\Omega}u\nabla|\nabla
v|^{2k-2}\cdot\nabla v-\int_{\Omega}u|\nabla v|^{2k-2}\Delta v$ (3.24)
$\displaystyle=-2(k-1)\int_{\Omega}u|\nabla v|^{2k-4}\nabla
v\cdot(D^{2}v\cdot\nabla v)-\int_{\Omega}u|\nabla v|^{2k-2}\Delta v$
$\displaystyle\leq(2(k-1)+\sqrt{n})\int_{\Omega}u|\nabla v|^{2k-2}|D^{2}v|.$

A substitution of (3.22), (3.23) and (3.1) into (3.1) shows (3.20). ∎

Next, we use the $L^{\infty}$-bound of $v$ provided by Lemma 3.1 to control
the terms on the right-hand side of (3.20).

######  Lemma 3.4.

For $k\geq 1$, the local-in-time classical solution of (1.4) obtained in Lemma
2.1 satisfies, for $t\in[0,T_{\text{max}})$, that

$\begin{split}&\frac{d}{dt}\int_{\Omega}|\nabla v|^{2k}+2k\int_{\Omega}|\nabla
v|^{2k}+\frac{k}{2}(3d-d_{\Omega})\int_{\Omega}|\nabla
v|^{2k-2}|D^{2}v|^{2}\\\ &\ \ \ +k(k-1)(d-d_{\Omega})\int_{\Omega}|\nabla
v|^{2k-4}|\nabla|\nabla v|^{2}|^{2}\\\
&\leq\frac{C_{1}}{d^{k}}(1+M_{0}\xi_{2})^{k+1}M_{0}^{k-1}\int_{\Omega}u^{k+1}+C_{1}d_{\Omega}M_{0}^{2k},\end{split}$
(3.25)

where $C_{1}>0$ depends only on $n,k,u_{0},v_{0}$ and $\Omega$ and
$d_{\Omega}$ is defined by (1.10).

###### Proof.

By (2.8) and (2.10) and the fact $\|v(\cdot,t)\|_{L^{\infty}}\leq M_{0}$ in
(3.1), one obtains that

$\displaystyle J_{1}:$ $\displaystyle=2k\xi_{2}\int_{\Omega}|\nabla
v|^{2k-2}\nabla v\cdot\nabla(\nabla v\cdot\nabla w)$ $\displaystyle\leq
2k\xi_{2}(\frac{\sqrt{n}}{2k}+1)\|\nabla
v\|^{2k}_{L^{2(k+1)}}\|D^{2}w\|_{L^{k+1}}$
$\displaystyle=2k\xi_{2}(\frac{\sqrt{n}}{2k}+1)\left(\int_{\Omega}|\nabla
v|^{2(k+1)}\right)^{\frac{k}{k+1}}\|D^{2}w\|_{L^{k+1}}$ (3.26)
$\displaystyle\leq
2k\xi_{2}(\frac{\sqrt{n}}{2k}+1)(2k+\sqrt{n})^{\frac{2k}{k+1}}\|v\|^{\frac{2k}{k+1}}_{L^{\infty}}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\|D^{2}w\|_{L^{k+1}}$
$\displaystyle\leq
2k\xi_{2}(\frac{\sqrt{n}}{2k}+1)(2k+\sqrt{n})^{\frac{2k}{k+1}}M_{0}^{\frac{2k}{k+1}}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\|D^{2}w\|_{L^{k+1}}$

for all $t\in(0,T_{max})$. On the other hand, by the uniqueness of the
elliptic problem

$-\Delta w=u-\bar{u}\text{ in }\Omega,\ \ \ \ \frac{\partial
w}{\partial\nu}=0\text{ on }\partial\Omega,\ \ \ \ \int_{\Omega}w=0,$ (3.27)

the well-known $W^{2,p}$-elliptic estimate (cf. [1, 2, 19], indeed,
$\Delta^{-1}:L^{p}\rightarrow W^{2,p}$ is a homeomorphism) shows that

$\|\Delta w\|_{L^{k+1}}\leq\sqrt{n}\|D^{2}w\|_{L^{k+1}}\leq
c_{4}\sqrt{n}\|u-\bar{u}\|_{L^{k+1}}\leq
2c_{4}\sqrt{n}\|u\|_{L^{k+1}}:=c_{5}\|u\|_{L^{k+1}},$ (3.28)

where $c_{5}$ depends only on $n$ and $\Omega$. Then we substitute (3.28) into
(3.1) to get

$J_{1}\leq
2c_{5}k\xi_{2}(\frac{1}{2k}+\frac{1}{\sqrt{n}})(2k+\sqrt{n})^{\frac{2k}{k+1}}{M_{0}}^{\frac{2k}{k+1}}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\|u\|_{L^{k+1}}.$ (3.29)

Furthermore, using (2.9), (2.10) and the fact (3.28), one can derive

$\begin{split}J_{2}&:=-2k\xi_{2}\int_{\Omega}v\Delta w\nabla\cdot(|\nabla
v|^{2k-2}\nabla v)\\\ &\leq
2k\xi_{2}(2(k-1)+\sqrt{n})\|v\|_{L^{\infty}}\|\nabla
v\|_{L^{2(k+1)}}^{k-1}\left(\int_{\Omega}|\nabla
v|^{2k-2}|D^{2}v|^{2}\right)^{\frac{1}{2}}\|\Delta w\|_{L^{k+1}}\\\ &\leq
2k\xi_{2}(2(k-1)+\sqrt{n})(2k+\sqrt{n})^{\frac{k-1}{k+1}}\|v\|_{L^{\infty}}^{\frac{2k}{k+1}}\left(\int_{\Omega}|\nabla
v|^{2k-2}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\|\Delta w\|_{L^{k+1}}\\\ &\leq
2c_{5}k\xi_{2}(2(k-1)+\sqrt{n})(2k+\sqrt{n})^{\frac{k-1}{k+1}}{M_{0}}^{\frac{2k}{k+1}}\left(\int_{\Omega}|\nabla
v|^{2k-2}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\|u\|_{L^{k+1}}.\end{split}$
(3.30)

In addition, applying the Hölder’s inequality together with (2.10), we have

$\begin{split}J_{3}:&=2k(2(k-1)+\sqrt{n})\int_{\Omega}u|\nabla
v|^{2k-2}|D^{2}v|\\\ &\leq 2k(2(k-1)+\sqrt{n})\left(\int_{\Omega}u^{2}|\nabla
v|^{2(k-1)}\right)^{\frac{1}{2}}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{1}{2}}\\\ &\leq
2k(2(k-1)+\sqrt{n})\|u\|_{L^{k+1}}\|\nabla
v\|_{L^{2(k+1)}}^{k-1}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{1}{2}}\\\ &\leq
2k(2(k-1)+\sqrt{n})(2k+\sqrt{n})^{\frac{k-1}{k+1}}{M_{0}}^{\frac{k-1}{k+1}}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\cdot\|u\|_{L^{k+1}}.\end{split}$
(3.31)

Now, summing over (3.29), (3.30) and (3.31), we infer

$J_{1}+J_{2}+J_{3}\leq c_{6}\left(\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}\right)^{\frac{k}{k+1}}\|u\|_{L^{k+1}},$ (3.32)

where

$\begin{split}c_{6}&=2c_{5}k\xi_{2}(\frac{1}{2k}+\frac{1}{\sqrt{n}})(2k+\sqrt{n})^{\frac{2k}{k+1}}{M_{0}}^{\frac{2k}{k+1}}\\\
&\ \
+2c_{5}k\xi_{2}(2(k-1)+\sqrt{n})(2k+\sqrt{n})^{\frac{k-1}{k+1}}{M_{0}}^{\frac{2k}{k+1}}\\\
&\ \
+2k(2(k-1)+\sqrt{n})(2k+\sqrt{n})^{\frac{k-1}{k+1}}{M_{0}}^{\frac{k-1}{k+1}}\\\
&\leq
6c_{5}k\xi_{2}(2k+\sqrt{n})^{\frac{2k}{k+1}}M_{0}^{\frac{2k}{k+1}}+2k(2k+\sqrt{n})^{\frac{2k}{k+1}}M_{0}^{\frac{k-1}{k+1}}\\\
&=2k(1+3c_{5}\xi_{2}M_{0})(2k+\sqrt{n})^{\frac{2k}{k+1}}M_{0}^{\frac{k-1}{k+1}}.\end{split}$
(3.33)

Therefore, by means of the Young’s inequality, we infer from (3.32) and (3.33)
that

$\begin{split}&J_{1}+J_{2}+J_{3}\\\ &\leq\frac{k}{2}d\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}+\frac{c_{6}^{k+1}}{k+1}\left(\frac{2}{d(k+1)}\right)^{k}\int_{\Omega}u^{k+1}\\\
&\leq\frac{k}{2}d\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}+\frac{2^{2k+1}}{d^{k}}(1+3c_{5}\xi_{2}M_{0})^{k+1}(2k+\sqrt{n})^{2k}M_{0}^{k-1}\int_{\Omega}u^{k+1}.\end{split}$
(3.34)

To control the boundary integral on (3.20), we first state the following well-
known fact due to homogeneous Neumann conditions:

$\frac{\partial|\nabla v|^{2}}{\partial\nu}\leq 2\sigma_{\Omega}|\nabla
v|^{2}\text{ on }\partial\Omega,$ (3.35)

where $\sigma_{\Omega}=\sigma 1_{\Omega}$ with $\sigma$ being the maximum
curvature of $\partial\Omega$ and $1_{\Omega}$ being the indicator whether
$\Omega$ is non-convex defined by

$1_{\Omega}=\begin{cases}0,&\text{ if }\Omega\text{ is convex
\cite[cite]{[\@@bibref{Number}{Ma-79}{}{}, Lemma 5.3]}},\\\ 1,&\text{ if
}\Omega\text{ is non-convex \cite[cite]{[\@@bibref{Number}{MS-
Poincare-2014}{}{}, Lemma 4.2]}}.\end{cases}$

Now, combining (3.35) and using the trace inequality
$\|\psi\|_{L^{2}(\partial\Omega)}\leq\varepsilon\|\nabla\psi\|_{L^{2}(\Omega)}+C_{\varepsilon}\|\psi\|_{L^{2}(\Omega)}$
for any $\varepsilon>0$ (cf. [36, Remark 52.9] and [44, (3.19)]), we have

$\begin{split}kd\int_{\partial\Omega}|\nabla v|^{2k-2}\frac{\partial|\nabla
v|^{2}}{\partial\nu}&\leq 2kd\sigma_{\Omega}\||\nabla
v|^{k}\|_{L^{2}(\partial\Omega)}^{2}\\\ &\leq
k(k-1)d_{\Omega}\int_{\Omega}|\nabla v|^{2(k-2)}|\nabla|\nabla
v|^{2}|^{2}+c_{7}d_{\Omega}\int_{\Omega}|\nabla v|^{2k},\end{split}$ (3.36)

where $d_{\Omega}$ is defined by (1.10). By (2.10), Young’s inequality with
$\varepsilon$ and the fact $\|v(\cdot,t)\|_{L^{\infty}}\leq M_{0}$ in (3.1),
we infer that

$\begin{split}c_{7}d_{\Omega}\int_{\Omega}|\nabla
v|^{2k}&\leq\frac{kd_{\Omega}}{2\left(2k+\sqrt{n}\right)^{2}M_{0}^{2}}\int_{\Omega}|\nabla
v|^{2(k+1)}+c_{8}d_{\Omega}M_{0}^{2k}\\\
&\leq\frac{k}{2}d_{\Omega}\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}+c_{8}d_{\Omega}M_{0}^{2k}.\end{split}$ (3.37)

Finally, substituting (3.34), (3.36) and (3.37) into (3.20) and then keeping
key parameters like $\xi_{2}$ and $M_{0}$, we accomplish our desired estimate
(3.25). ∎

######  Lemma 3.5.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 1)$ and $(u,v,w)$ be the solution of
(1.4) obtained in Lemma 2.1. Then, for any $k\geq 1$, there exist two positive
constants $\xi_{*}(k)$ and $\mu_{*}(k)$ defined respectively by (3.52) and
(3.57) such that whenever one of the following conditions holds:

$(1)\ \xi_{1}\geq\xi_{*}(k)\chi^{2};\ \ \ (2)\ \theta=1,\
\mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{*}(k)\chi^{\frac{2}{5+n}};\ \ \ (3)\
\theta>1,\ \mu>0,$ (3.38)

there exist three positive constants $C_{2},C_{3},C_{4}$ independent of
$t,\chi,\mu,d$ and $\xi_{i}~{}(i=1,2)$ but depending on
$u_{0},v_{0},\theta,a,k,n$ and $\Omega$ such that, for
$t\in(0,T_{\text{max}})$,

$\int_{\Omega}u^{k}\leq
C_{2}\left(1+m_{1}^{k}+M_{1}(k)+(1+d_{\Omega}M_{0}^{2k})d^{-1}\chi^{2}M_{0}^{2-k}+M_{2}^{c}(k)\right):=C_{2}M_{2}(k),$
(3.39)

and

$\int_{\Omega}|\nabla
v|^{2k}\leq\frac{C_{3}M_{2}(k)}{\color[rgb]{0,0,0}{(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}}},$
(3.40)

as well as, if $k>n$,

$\|w(\cdot,t)\|_{W^{1,\infty}}\leq C_{4}M_{2}^{\frac{1}{k}}(k),$ (3.41)

where $M_{2}^{c}$ is defined by

$M_{2}^{c}(k):=\begin{cases}0,\quad\quad\quad\quad\quad\quad\quad\text{if
}\xi_{1}\geq\xi_{*}(k)\chi^{2},\\\\[7.11317pt]
0,\quad\quad\quad\quad\quad\quad\theta=1,\text{ if }\mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{*}(k)\chi^{\frac{2}{5+n}},\\\\[7.11317pt]
\frac{(\theta-1)}{\mu^{\frac{k+1}{\theta-1}}}\left[(1+\frac{1}{d^{k+1}})(1+M_{0}\xi_{2})M_{0}\chi^{2}\right]^{\frac{k+\theta}{\theta-1}},\
\text{if }\theta>1,~{}\mu>0.\end{cases}$ (3.42)

###### Proof.

We apply the estimate (2.10) and the fact $\|v(\cdot,t)\|_{L^{\infty}}\leq
M_{0}$ to deduce that

$\begin{split}\frac{\chi^{2}(k-1)}{2}\int_{\Omega}u^{k}|\nabla
v|^{2}&\leq\varepsilon_{1}\int_{\Omega}u^{k+1}+c_{9}\chi^{2(k+1)}\varepsilon_{1}^{-k}\int_{\Omega}|\nabla
v|^{2(k+1)}\\\
&\leq\varepsilon_{1}\int_{\Omega}u^{k+1}+c_{10}\chi^{2(k+1)}M_{0}^{2}\varepsilon_{1}^{-k}\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2},\\\ \end{split}$ (3.43)

where $\varepsilon_{1}>0$, to be chosen in (3.47) below. On the one hand,
using the widely known Gagliardo-Nirenberg inequality (cf. [39, Section 3]),
we can find a constant $c_{11}>0$ only depending on $k,n$ and $\Omega$ and
$\alpha=\frac{\frac{k}{2}-\frac{1}{2}}{\frac{k}{2}-\frac{1}{2}+\frac{1}{n}}\in(0,1)$
such that

$\begin{split}\int_{\Omega}u^{k}=\|u^{\frac{k}{2}}\|^{2}_{L^{2}}&\leq
c_{11}(\|\nabla
u^{\frac{k}{2}}\|^{2\alpha}_{L^{2}}\|u^{\frac{k}{2}}\|^{2(1-\alpha)}_{L^{\frac{2}{k}}}+\|u^{\frac{k}{2}}\|^{2}_{L^{\frac{2}{k}}})\\\
&\leq c_{11}\left(\frac{k^{2}}{4}\int_{\Omega}u^{k-2}|\nabla
u|^{2}\right)^{\alpha}m_{1}^{k(1-\alpha)}+c_{11}m_{1}^{k}\\\
&\leq\frac{k(k-1)}{2}\int_{\Omega}u^{k-2}|\nabla
u|^{2}+c_{12}m_{1}^{k}.\end{split}$ (3.44)

Then substituting (3.43) and (3.44) into (3.14) with $p=k$, one has

$\begin{split}&\frac{d}{dt}\int_{\Omega}u^{k}+{\color[rgb]{0,0,0}\int_{\Omega}u^{k}}+\frac{\xi_{1}(k-1)}{\color[rgb]{0,0,0}2}\int_{\Omega}u^{k+1}+\frac{k\mu}{2}\int_{\Omega}u^{\theta+k}\\\
&\leq\varepsilon_{1}\int_{\Omega}u^{k+1}+c_{10}\chi^{2(k+1)}M_{0}^{2}\varepsilon_{1}^{-k}\int_{\Omega}|\nabla
v|^{2(k-1)}|D^{2}v|^{2}+c_{12}m_{1}^{k}+M_{1}(k).\end{split}$ (3.45)

On the other hand, it follows from Lemma 3.4 and the fact $d_{\Omega}\leq d$
that

$\begin{split}&\frac{d}{dt}\int_{\Omega}|\nabla v|^{2k}+2k\int_{\Omega}|\nabla
v|^{2k}+kd\int_{\Omega}|\nabla v|^{2k-2}|D^{2}v|^{2}\\\
&\leq\frac{C_{1}}{d^{k}}(1+M_{0}\xi_{2})^{k+1}M_{0}^{k-1}\int_{\Omega}u^{k+1}+C_{1}d_{\Omega}M_{0}^{2k}.\end{split}$
(3.46)

Let us first set

$c_{13}=\frac{C_{1}c_{10}}{k},\ \ \ \
\varepsilon_{1}=(1+M_{0}\xi_{2})M_{0}\chi^{2}(kc_{13})^{\frac{1}{k+1}},$
(3.47)

and then we put

$\delta(\varepsilon_{1})=\frac{c_{10}\chi^{2(k+1)}M_{0}^{2}\varepsilon_{1}^{-k}}{kd}=\frac{c_{10}}{k}(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}(kc_{13})^{-\frac{k}{k+1}},$
(3.48)

and

$y(t):=\int_{\Omega}u^{k}+\delta(\varepsilon_{1})\int_{\Omega}|\nabla
v|^{2k}.$ (3.49)

Then we multiply (3.46) by $\delta(\varepsilon_{1})$ and add the result to
(3.45) to obtain

$\begin{split}&\ \ \ \
y^{\prime}(t)+y(t)+\frac{\xi_{1}(k-1)}{2}\int_{\Omega}u^{k+1}+\frac{k\mu}{2}\int_{\Omega}u^{\theta+k}\\\
&\leq\left\\{c_{13}\left(\frac{1}{d}(1+M_{0}\xi_{2})M_{0}\chi^{2}\right)^{k+1}\varepsilon_{1}^{-k}+\varepsilon_{1}\right\\}\int_{\Omega}u^{k+1}+c_{14}\\\
&=(kc_{13})^{\frac{1}{k+1}}\left(1+\frac{1}{kd^{k+1}}\right)(1+M_{0}\xi_{2})M_{0}\chi^{2}\int_{\Omega}u^{k+1}+c_{14},\\\
\end{split}$ (3.50)

where

$c_{14}=c_{12}m_{1}^{k}+M_{1}(k)+\delta(\varepsilon_{1})C_{1}d_{\Omega}M_{0}^{2k}.$
(3.51)

To show that the first term on the right-hand side of (3.50) can be absorbed
by the terms involving $\int_{\Omega}u^{k+1}$ or $\int_{\Omega}u^{\theta+k}$,
we first define

$\xi_{*}(k)=\frac{2(kc_{13})^{\frac{1}{k+1}}}{k-1}\left(1+\frac{1}{kd^{k+1}}\right)(1+M_{0}\xi_{2})M_{0}.$
(3.52)

Case 1: $\xi_{1}\geq\xi_{*}(k)\chi^{2}$. In this case, notice that $M_{0}$ is
independent of $\xi_{1}$ by (3.1) and (2.2), in view of (3.50) along with
(3.51) and (3.52), we deduce that

$y^{\prime}(t)+y(t)\leq
c_{12}m_{1}^{k}+M_{1}(k)+\delta(\varepsilon_{1})C_{1}d_{\Omega}M_{0}^{2k}.$
(3.53)

Solving this ODI and using (3.48) and (3.49), we get that

$\begin{split}&\int_{\Omega}u^{k}+\frac{c_{10}}{k}(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}(kc_{13})^{-\frac{k}{k+1}}\int_{\Omega}|\nabla
v|^{2k}\\\\[7.11317pt]
&\leq\|u_{0}\|^{k}_{L^{k}}+c_{15}(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\|\nabla
v_{0}\|^{2k}_{L^{2k}}\\\\[7.11317pt] &\ \
+c_{12}m_{1}^{k}+M_{1}(k)+c_{15}(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{k+2}d_{\Omega}\\\\[7.11317pt]
&\leq
c_{16}\left[1+m_{1}^{k}+M_{1}(k)+(1+d_{\Omega}M_{0}^{2k})d^{-1}\chi^{2}M_{0}^{2-k}\right].\end{split}$
(3.54)

Case 2: $\theta=1$ and $\mu>0$ is suitably large. In this case, we first
observe from (3.1) and (2.2) that $M_{0}$ is bounded by
$O(1)(1+\mu^{-(2+n/2)})$, and therefore,

$(1+M_{0}\xi_{2})M_{0}\leq
O(1)\left(1+\left(\frac{1}{\mu}\right)^{4+n}\right).$

This enables us to infer that

$\hat{\mu}_{*}(k)=\frac{2(kc_{13})^{\frac{1}{k+1}}}{k}\left(1+\frac{1}{kd^{k+1}}\right)\sup_{0<\mu<1}\left\\{\mu^{4+n}(1+M_{0}\xi_{2})M_{0}\right\\}<+\infty$
(3.55)

and

$\tilde{\mu}_{*}(k)=\frac{2(kc_{13})^{\frac{1}{k+1}}}{k}\left(1+\frac{1}{kd^{k+1}}\right)\sup_{\mu\geq
1}\left\\{(1+M_{0}\xi_{2})M_{0}\right\\}<+\infty.$ (3.56)

Now, we define

$\mu_{*}(k)=\max\left\\{\left(\hat{\mu}_{*}(k)\right)^{\frac{1}{5+n}},\ \ \
\tilde{\mu}_{*}(k)\right\\}<+\infty.$ (3.57)

Then, under (2) of (3.38), we see from (3.57) that

$\mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{*}(k)\chi^{\frac{2}{5+n}}\geq\max\left\\{\left(\hat{\mu}_{*}(k)\right)^{\frac{1}{5+n}}\chi^{\frac{2}{5+n}},\
\ \tilde{\mu}_{*}(k)\chi^{2}\right\\},$

and so (3.55) together with (3.56) implies

$\mu\geq\frac{2(kc_{13})^{\frac{1}{k+1}}}{k}\left(1+\frac{1}{kd^{k+1}}\right)(1+M_{0}\xi_{2})M_{0}\chi^{2}.$
(3.58)

Then in light of (3.50) along with (3.51) and (3.58), we derive an identical
ODI as (3.53), and then we get the same estimate as (3.54).

Case 3: $\theta>1$ and $\mu>0$. Let

$\begin{split}c_{17}&:=\sup_{s>0}\left\\{(kc_{13})^{\frac{1}{k+1}}\left(1+\frac{1}{kd^{k+1}}\right)(1+M_{0}\xi_{2})M_{0}\chi^{2}s^{k+1}-\frac{k\mu}{2}s^{k+\theta}\right\\}\\\
&=\frac{(\theta-1)}{k+\theta}\left(\frac{2(k+1)}{k(k+\theta)\mu}\right)^{\frac{k+1}{\theta-1}}\left[(kc_{13})^{\frac{1}{k+1}}\left(1+\frac{1}{kd^{k+1}}\right)(1+M_{0}\xi_{2})M_{0}\chi^{2}\right]^{\frac{k+\theta}{\theta-1}}.\end{split}$
(3.59)

Then, combining (3.59) and (3.50) along with (3.51), we end up with

$y^{\prime}(t)+y(t)\leq
c_{12}m_{1}^{k}+M_{1}(k)+\delta(\varepsilon_{1})C_{1}d_{\Omega}M_{0}^{2k}+c_{17}|\Omega|,$

which in conjunction with (3.54), (3.59) and (3.48) allows us to deduce that

$\begin{split}&\int_{\Omega}u^{k}+\frac{c_{10}}{k}(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}(kc_{13})^{-\frac{k}{k+1}}\int_{\Omega}|\nabla
v|^{2k}\\\\[7.11317pt] &\leq
c_{16}\left[1+m_{1}^{k}+M_{1}(k)+(1+d_{\Omega}M_{0}^{2k})d^{-1}\chi^{2}M_{0}^{2-k}\right]\\\\[7.11317pt]
&\ \ \
+c_{18}\frac{(\theta-1)}{\mu^{\frac{k+1}{\theta-1}}}\left[\left(1+\frac{1}{d^{k+1}}\right)(1+M_{0}\xi_{2})M_{0}\chi^{2}\right]^{\frac{k+\theta}{\theta-1}}.\end{split}$
(3.60)

The desired qualitative $(L^{k},L^{2k})$-bounds of $(u,\nabla v)$ in (3.39)
and (3.40) are resulted from (3.54) and (3.60) upon simple algebraic
manipulations. Finally, if $k>n$, then the $W^{2,k}$-estimate (3.28) and the
Sobolev embedding $W^{2,k}\hookrightarrow W^{1,\infty}$ together imply (3.41).
∎

###  3.2. Qualitative $(L^{\infty},W^{1,\infty})$-boundedness of $(u,v)$

Before proceeding, based on (3.52) and (3.57), we first define

$\xi_{0}=\inf_{k>\frac{n}{2}}\xi_{*}(k)=\inf_{k>\frac{n}{2}}\left\\{\frac{2(kc_{13})^{\frac{1}{k+1}}}{k-1}\left(1+\frac{1}{kd^{k+1}}\right)\right\\}(1+M_{0}\xi_{2})M_{0}$
(3.61)

and, with $\hat{\mu}_{*}(k)$ and $\tilde{\mu}_{*}(k)$ defined by (3.55) and
(3.56),

$\mu_{0}=\inf_{k>\frac{n}{2}}\mu_{*}(k)=\inf_{k>\frac{n}{2}}\left\\{\max\left\\{\left(\hat{\mu}_{*}(k)\right)^{\frac{1}{5+n}},\
\ \ \tilde{\mu}_{*}(k)\right\\}\right\\}.$ (3.62)

######  Lemma 3.6.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ and $(u,v,w)$ be the local solution
of (1.4) obtained in Lemma 2.1. Suppose that one of the following conditions
holds:

$(1)\ \xi_{1}>\xi_{0}\chi^{2};\ \ (2)\ \theta=1,\ \mu>\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{0}\chi^{\frac{2}{5+n}};\ \ (3)\
\theta>1,\ \mu>0.$ (3.63)

Then the $L^{\infty}$-norm of the $u$-solution component is uniformly bounded
on $(0,T_{\text{max}})$. In particular, if one of the following conditions
holds:

$(1)\ \xi_{1}\geq\xi_{*}(3n)\chi^{2};\ \ (2)\ \theta=1,\
\mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{*}(3n)\chi^{\frac{2}{5+n}};\ \ (3)\
\theta>1,\mu>0,$ (3.64)

then there exists $C_{5}>0$ independent of
$t,\chi,\mu,d~{}and~{}\xi_{i}(i=1,2)$ but depending on $n,u_{0},v_{0}$ and
$\Omega$ such that the local solution $(u,v,w)$ of (1.4) obtained in Lemma 2.1
satisfies, for $t\in(0,T_{\text{max}})$, that

$\begin{split}&\|u(t)\|_{L^{\infty}}\leq C_{5}M_{3}(3n)\\\
&:=C_{5}\begin{cases}1+M_{2}^{\frac{1}{3n}}(3n)+\xi_{1}M_{2}^{\frac{2}{3n}}(3n)+\frac{\chi
M_{2}^{\frac{2}{3n}}(3n)}{\left((1+M_{0}\xi_{2})^{-3n}d^{-1}\chi^{2}M_{0}^{2-3n}\right)^{\frac{1}{3n}}},&\text{if
}a=\mu=0,\\\\[7.11317pt]
1+\left(\frac{1}{\mu}\right)^{\frac{1}{\theta}}+\xi_{1}M_{2}^{\frac{2}{3n}}(3n)+\frac{\chi
M_{2}^{\frac{2}{3n}}(3n)}{\left((1+M_{0}\xi_{2})^{-3n}d^{-1}\chi^{2}M_{0}^{2-3n}\right)^{\frac{1}{3n}}},&\text{if
}a,\mu>0.\end{cases}\end{split}$ (3.65)

where $M_{2}(\cdot)$ is defined by (3.42).

###### Proof.

By the definitions of $\xi_{0}$ and $\mu_{0}$ respectively in (3.61) and
(3.62), it is easy to see that the corresponding case of (3.63) implies that
of (3.38) for some $k>\frac{n}{2}$. Then an application of Lemma 3.5 shows

$\|u(\cdot,t)\|_{L^{k}}\leq c_{19}M_{2}^{\frac{1}{k}}(k),\quad\quad\|\nabla
v(\cdot,t)\|_{L^{2k}}\leq
c_{20}\left(\frac{M_{2}(k)}{(1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}}\right)^{\frac{1}{2k}}.$
(3.66)

Therefore, the $W^{2,k}$-estimate (3.28) yields

$\|w(\cdot,t)\|_{W^{2,k}}\leq c_{20}\|u(\cdot,t)-\bar{u}\|_{L^{k}}\leq
2c_{20}\|u(\cdot,t)\|_{L^{k}}\leq 2c_{19}c_{20}M_{2}^{\frac{1}{k}}(k),$

and so the Sobolev embedding $W^{2,k}\hookrightarrow W^{1,q}$ for some $q>n$
shows

$\|w(\cdot,t)\|_{W^{1,q}}\leq c_{21}M_{2}^{\frac{1}{k}}(k),\quad\quad
q=\frac{4nk}{k+3(n-k)^{+}}.$ (3.67)

We now use similar sprits as used in [16, 44] to derive the $L^{\infty}$-bound
of $u$. To that purpose, we use the variation-of-constants formula to the
$u$-equation in (1.4) to write

$\begin{split}u(\cdot,t)&=e^{t(\Delta-1)}u_{0}+\int_{0}^{t}e^{(\Delta-1)(t-s)}\nabla\cdot\\{u(\cdot,s)(\xi_{1}\nabla
w(\cdot,s)-\chi\nabla v(\cdot,s))\\}ds\\\
&\;\;\;+\int_{0}^{t}e^{(\Delta-1)(t-s)}\left[(a+1)u(\cdot,s)-\mu
u^{\theta+1}(\cdot,s)\right]ds\\\
&=:u_{1}(\cdot,t)+u_{2}(\cdot,t)+u_{3}(\cdot,t).\end{split}$

We next estimate $u_{1},u_{2}$ and $u_{3}$. First, the nonnegativity of $u$
shows, for $t\in(0,T_{\text{max}})$,

$\|u(\cdot,t)\|_{L^{\infty}}=\sup_{x\in\Omega}u(x,t)\leq\sup_{x\in\Omega}u_{1}(x,t)+\sup_{x\in\Omega}u_{2}(x,t)+\sup_{x\in\Omega}u_{3}(x,t).$
(3.68)

By the order property of the Neumann heat semigroup $(e^{t\Delta})_{t>0}$ due
to the maximum principle, we estimate $u_{1}$ and $u_{3}$ in the following
ways:

$\|u_{1}(\cdot,t)\|_{L^{\infty}}=\|e^{t(\Delta-1)}u_{0}\|_{L^{\infty}}\leq
e^{-t}\|u_{0}\|_{L^{\infty}}\leq\|u_{0}\|_{L^{\infty}},$ (3.69)

and, the semigroup estimate (2.13) in Lemma 2.5 and (3.66) yield

$\begin{split}u_{3}(\cdot,t)&=\int_{0}^{t}e^{(\Delta-1)(t-s)}\left[(a+1)u(\cdot,s)-\mu
u^{\theta+1}(\cdot,s)\right]ds\\\\[7.11317pt]
&\leq\begin{cases}\int_{0}^{t}e^{(\Delta-1)(t-s)}u(\cdot,s)ds,&\text{ if
}a=\mu=0,\\\\[7.11317pt]
\int_{0}^{t}e^{(\Delta-1)(t-s)}\left(\frac{a+1}{(1+\theta)\mu}\right)^{\frac{1}{\theta}}\frac{(a+1)\theta}{1+\theta}ds,&\text{
if }a\geq 0,\ \ \mu>0,\end{cases}\\\\[7.11317pt] &\leq
c_{21}\begin{cases}M_{2}^{\frac{1}{k}}(k)\int_{0}^{t}(1+(t-s)^{-\frac{n}{2k}})e^{-(t-s)}ds,&\text{
if }a=\mu=0,\\\\[7.11317pt]
\int_{0}^{t}e^{-(t-s)}e^{\Delta(t-s)}\left(\frac{1}{\mu}\right)^{\frac{1}{\theta}}ds,&\text{
if }a\geq 0,\ \ \mu>0,\end{cases}\\\\[7.11317pt] &\leq
c_{22}\begin{cases}M_{2}^{\frac{1}{k}}(k),&\text{ if }a=\mu=0,\\\\[7.11317pt]
\left(\frac{1}{\mu}\right)^{\frac{1}{\theta}},&\text{ if }a\geq 0,\ \
\mu>0.\end{cases}\end{split}$ (3.70)

For convenience of reference, we list two Hölder’s type interpolation
inequalities:

$\|fg\|_{L^{r}}\leq\|f\|_{L^{p}}\|g\|_{L^{q}}\text{ with }p,q,r\geq 1,\ \
\frac{1}{r}=\frac{1}{p}+\frac{1}{q},$

and

$\|f\|_{L^{r}}\leq\|f\|_{L^{\infty}}^{\frac{(r-k)^{+}}{r}}\|f\|_{L^{k}}^{\frac{k}{k+(r-k)^{+}}}|\Omega|^{\frac{(k-r)^{+}}{kr}}\text{
with }k,r\geq 1.$

Employing the semigroup estimate (2.14) in Lemma 2.5, Hölder’s interpolation
inequalities above and (3.67), we deduce, for $k>\frac{n}{2}$, that

$\begin{split}&\left\|\xi_{1}\int_{0}^{t}e^{(\Delta-1)(t-s)}\nabla\cdot(u\nabla
w)ds\right\|_{L^{\infty}}\\\ &\leq
c_{23}\xi_{1}\int_{0}^{t}\left(1+(t-s)^{-1+\frac{k-(n-k)^{+}}{8k}}\right)e^{-(t-s)}\|u\nabla
w\|_{L^{\frac{4nk}{3k+(n-k)^{+}}}}ds\\\ &\leq
c_{24}\xi_{1}\int_{0}^{t}\left(1+(t-s)^{-1+\frac{k-(n-k)^{+}}{8k}}\right)e^{-(t-s)}\|u\|_{L^{\frac{2nk}{k-(n-k)^{+}}}}\|\nabla
w\|_{L^{\frac{4nk}{k+3(n-k)^{+}}}}ds\\\ &\leq
c_{25}\xi_{1}M_{2}^{\frac{2k-2(n-k)^{+}+\left(2n-k+(n-k)^{+}\right)^{+}}{\left[k-(n-k)^{+}+\left(2n-k+(n-k)^{+}\right)^{+}\right]k}}(k)\left(\sup_{s\in(0,t)}\|u(s)\|_{L^{\infty}}\right)^{\frac{\left(2n-k+(n-k)^{+}\right)^{+}}{2n}}\\\
&=c_{25}\xi_{1}\begin{cases}M_{2}^{\frac{2n+k-(n-k)^{+}}{2nk}}(k)\left(\sup_{s\in(0,t)}\|u(s)\|_{L^{\infty}}\right)^{\frac{2n-k+(n-k)^{+}}{2n}},&\text{
if }k<2n,\\\\[7.11317pt] M_{2}^{\frac{2}{k}}(k),&\text{ if }k\geq
2n,\end{cases}\\\
&\leq\begin{cases}\frac{1}{4}\sup_{s\in(0,t)}\|u(s)\|_{L^{\infty}}+c_{26}\xi_{1}^{\frac{2n}{k-(n-k)^{+}}}M_{2}^{\frac{2n+k-(n-k)^{+}}{[k-(n-k)^{+}]k}}(k),&\text{
if }k<2n,\\\\[7.11317pt] c_{25}\xi_{1}M_{2}^{\frac{2}{k}}(k),&\text{ if }k\geq
2n,\end{cases}\end{split}$ (3.71)

where we used the fact that

$\displaystyle\|u\|_{L^{\frac{2nk}{k-(n-k)^{+}}}}\|\nabla
w\|_{L^{\frac{4nk}{k+3(n-k)^{+}}}}$ $\displaystyle\leq
c_{27}\|u\|_{L^{k}}^{\frac{k-(n-k)^{+}}{k-(n-k)^{+}+\left(2n-k+(n-k)^{+}\right)^{+}}}\|u\|_{L^{\infty}}^{\frac{\left(2n-k+(n-k)^{+}\right)^{+}}{2n}}\|\nabla
w\|_{L^{\frac{4nk}{k+3(n-k)^{+}}}}$ $\displaystyle\leq
c_{28}M_{2}^{\frac{2k-2(n-k)^{+}+\left(2n-k+(n-k)^{+}\right)^{+}}{\left[k-(n-k)^{+}+\left(2n-k+(n-k)^{+}\right)^{+}\right]k}}\|u\|_{L^{\infty}}^{\frac{\left(2n-k+(n-k)^{+}\right)^{+}}{2n}}$

and the finiteness of gamma integral due to the fact that $k-(n-k)^{+}>0$:

$\int_{0}^{t}\left(1+(t-s)^{-1+\frac{k-(n-k)^{+}}{8k}}\right)e^{-(t-s)}ds=\int_{0}^{t}\left(1+\tau^{-1+\frac{k-(n-k)^{+}}{8k}}\right)e^{-\tau}d\tau<\infty.$

Similarly, using the boundedness information in (3.66), we infer

$\begin{split}&\left\|-\chi\int_{0}^{t}e^{(\Delta-1)(t-s)}\nabla\cdot(u\nabla
v)(\cdot,s)ds\right\|_{L^{\infty}}\\\ &\leq
c_{29}\chi\int_{0}^{t}\left(1+(t-s)^{-1+\frac{2k-n}{8k}}\right)e^{-(t-s)}\|(u\nabla
v)(\cdot,s)\|_{L^{\frac{4nk}{2k+n}}}ds\\\ &\leq
c_{30}\chi\int_{0}^{t}\left(1+(t-s)^{-1+\frac{2k-n}{8k}}\right)e^{-(t-s)}\|u(\cdot,s)\|_{L^{\frac{4nk}{2k-n}}}\|\nabla
v(\cdot,s)\|_{L^{2k}}ds\\\ &\leq c_{31}\frac{\chi
M_{2}^{\frac{4k-2n+(5n-2k)^{+}}{k[2k-n+(5n-2k)^{+}]}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{1}{k}}}\left(\sup_{s\in(0,t)}\|u(\cdot,s)\|_{L^{\infty}}\right)^{\frac{(5n-2k)^{+}}{4n}}\\\
&\leq\begin{cases}\frac{1}{4}\sup_{s\in(0,t)}\|u(s)\|_{L^{\infty}}+c_{32}\frac{\chi^{\frac{4n}{2k-n}}M_{2}^{\frac{2k+3n}{k(2k-n)}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{4n}{k(2k-n)}}},&\text{
if }k<\frac{5n}{2},\\\\[7.11317pt] c_{31}\frac{\chi
M_{2}^{\frac{2}{k}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{1}{k}}},&\text{
if }k\geq\frac{5n}{2}.\end{cases}\end{split}$ (3.72)

In the case of $k\geq\frac{5n}{2}$, substituting (3.69), (3.70), (3.71) and
(3.72) into (3.68), we accomplish that

$\|u(t)\|_{L^{\infty}}\leq
c_{32}\begin{cases}1+M_{2}^{\frac{1}{k}}(k)+\xi_{1}M_{2}^{\frac{2}{k}}(k)+\frac{\chi
M_{2}^{\frac{2}{k}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{1}{k}}},&\text{if
}\mu=0,\\\\[7.11317pt]
1+\left(\frac{1}{\mu}\right)^{\frac{1}{\theta}}+\xi_{1}M_{2}^{\frac{2}{k}}(k)+\frac{\chi
M_{2}^{\frac{2}{k}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{1}{k}}},&\text{if
}\mu>0.\end{cases}$ (3.73)

Similarly, in the case of $k\in[2n,\frac{5n}{2})$, we have

$\begin{split}\|u(t)\|_{L^{\infty}}\leq
c_{33}\begin{cases}1+M_{2}^{\frac{1}{k}}(k)+\xi_{1}M_{2}^{\frac{2}{k}}(k)+\frac{\chi^{\frac{4n}{2k-n}}M_{2}^{\frac{2k+3n}{k(2k-n)}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{4n}{k(2k-n)}}},&\text{if
}\mu=0,\\\\[5.69046pt]
1+\left(\frac{1}{\mu}\right)^{\frac{1}{\theta}}+\xi_{1}M_{2}^{\frac{2}{k}}(k)+\frac{\chi^{\frac{4n}{2k-n}}M_{2}^{\frac{2k+3n}{k(2k-n)}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{4n}{k(2k-n)}}},&\text{if
}\mu>0.\end{cases}\end{split}$ (3.74)

And in the case of $k\in(\frac{n}{2},2n)$, we get

$\begin{split}\|u(t)\|_{L^{\infty}}\leq&c_{34}\left(1+\frac{\chi^{\frac{4n}{2k-n}}M_{2}^{\frac{2k+3n}{k(2k-n)}}(k)}{\left((1+M_{0}\xi_{2})^{-k}d^{-1}\chi^{2}M_{0}^{2-k}\right)^{\frac{4n}{k(2k-n)}}}\right)\\\
&+c_{34}\xi_{1}^{\frac{2n}{k-(n-k)^{+}}}M_{2}^{\frac{2n+k-(n-k)^{+}}{[k-(n-k)^{+}]k}}(k)+c_{34}\begin{cases}M_{2}^{\frac{1}{k}}(k),&\text{
if }\mu=0,\\\\[7.11317pt]
\left(\frac{1}{\mu}\right)^{\frac{1}{\theta}},&\text{ if
},\mu>0.\end{cases}\end{split}$ (3.75)

Combining (3.73), (3.74) and (3.75), we see either one of (3.63) implies that
$\|u(\cdot,t)\|_{L^{\infty}}$ is uniformly bounded for
$t\in(0,T_{\text{max}})$. In particular, if one of (3.64) holds, then the
desired qualitative bound of $u$ in (3.65) follows upon setting $k=3n$ in
(3.73). ∎

######  Lemma 3.7.

Let $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ and $(u,v,w)$ be the solution of
(1.4) obtained in Lemma 2.1. If one of (3.63) holds, then $\|\nabla
v(\cdot,t)\|_{L^{\infty}}$ is uniformly bounded on $(0,T_{\text{max}})$. In
particular, if one of the following conditions holds:

$(1)\ \xi_{1}\geq\xi_{*}(3n)\chi^{2};\ \ (2)\ \theta=1,\
\mu\geq\max\left\\{1,\
\chi^{\frac{8+2n}{5+n}}\right\\}\mu_{*}(3n)\chi^{\frac{2}{5+n}};\ \ (3)\
\theta>1,\mu>0,$ (3.76)

then, for $t\in(0,T_{\text{max}})$, there exists $C_{6}>0$ depending only on
$n,u_{0},v_{0}$ and $\Omega$ such that

$\begin{split}&\|\nabla v(\cdot,t)\|_{L^{\infty}}\\\ &\leq
C_{6}\left(1+\frac{\xi_{2}M_{2}^{\frac{1}{2n}}(3n)}{d\left((1+M_{0}\xi_{2})^{-3n}d^{-1}\chi^{2}M_{0}^{2-3n}\right)^{\frac{1}{6n}}}+\frac{\xi_{2}}{d}M_{0}M_{2}^{\frac{1}{3n}}(3n)+\frac{M_{2}^{\frac{1}{3n}}(3n)}{d}\right)\end{split}$
(3.77)

and

$\|w(\cdot,t)\|_{W^{2,\infty}}\leq C_{6}M_{3}(3n),$ (3.78)

where $M_{0}$, $M_{2}(\cdot)$ and $M_{3}(\cdot)$ are defined by (3.1), (3.39)
and (3.65) respectively.

###### Proof.

Using the time scaling $\tilde{t}=dt$, we rewrite the second equation in (1.4)
as follows:

$v_{\tilde{t}}=\Delta v-\frac{1}{d}v+\frac{\xi_{2}}{d}\nabla w\cdot\nabla
v+\frac{\xi_{2}}{d}v(\bar{u}-u)+\frac{1}{d}u.$

Then an application of the variation-of-constants formula yields

$\begin{split}v(\cdot,\tilde{t})&=e^{(\Delta-\frac{1}{d})\tilde{t}}v_{0}+\frac{\xi_{2}}{d}\int_{0}^{\tilde{t}}e^{(\Delta-\frac{1}{d})(\tilde{t}-s)}(\nabla
w\cdot\nabla v)(\cdot,s)ds\\\
&\;\;\;\;+\frac{\xi_{2}}{d}\int_{0}^{\tilde{t}}e^{(\Delta-\frac{1}{d})(\tilde{t}-s)}v(\cdot,s)(\bar{u}(s)-u(\cdot,s))ds+\frac{1}{d}\int_{0}^{\tilde{t}}e^{(\Delta-\frac{1}{d})(\tilde{t}-s)}u(\cdot,s)ds.\end{split}$
(3.79)

Under one of (3.63), we first see from Lemma 3.6 and its proof, for some
$k>\frac{n}{2}$, that

$\|u(\cdot,\tilde{t})\|_{L^{\infty}}+\|\nabla
v(\cdot,\tilde{t})\|_{L^{2k}}\leq c_{35},\quad\quad\forall\tilde{t}>0.$ (3.80)

Then the elliptic estimate applied to $\Delta w=\bar{u}-u$ along with Lemma
3.1 shows that

$\|v(\cdot,\tilde{t})\|_{L^{\infty}}+\|\nabla
w(\cdot,\tilde{t})\|_{L^{\infty}}\leq c_{36},\quad\quad\forall\tilde{t}>0.$
(3.81)

Then, with (3.80) and (3.81) at hand, we can apply the smoothing
$L^{p}$-$L^{q}$ semigroup estimates in Lemma 2.5 to (3.79) to infer that
$\|\nabla v(\cdot,t)\|_{L^{\infty}}$ is uniformly bounded. We next aim to
derive a qualitative bound for $\|\nabla v(\cdot,t)\|_{L^{\infty}}$ under one
of (3.76). In this circumstance, Lemma 3.5 allows us to see, for
$\tilde{t}\in(0,dT_{\text{max}})$, that

$\|u(\cdot,\tilde{t})\|_{L^{3n}}\leq c_{37}M_{2}^{\frac{1}{3n}}(3n),\ \ \
\|\nabla
v(\cdot,\tilde{t})\|_{L^{6n}}\leq\frac{c_{38}M_{2}^{\frac{1}{6n}}(3n)}{\left((1+M_{0}\xi_{2})^{-3n}d^{-1}\chi^{2}M_{0}^{2-3n}\right)^{\frac{1}{6n}}}.$
(3.82)

Hence, applying the $W^{2,\infty}$-estimate to (3.27) and using the
qualitative bound for $\|u\|_{L^{\infty}}$ in (3.65), we find that

$\|w(\cdot,\tilde{t})\|_{W^{2,\infty}}\leq c_{39}M_{3}(3n),\ \
\tilde{t}\in(0,dT_{\text{max}}).$

This directly entails (3.78).

Next, with the aid of (3.82) and (3.41) with $k=3n$, and keeping in mind that
$\|v(\cdot,\tilde{t})\|_{L^{\infty}}\leq M_{0}$ by Lemma 3.1, we utilize the
Hölder inequality and the semigroup estimates in Lemma 2.5 to (3.79) to deduce
that

$\displaystyle\|\nabla v(\cdot,\tilde{t})\|_{L^{\infty}}$
$\displaystyle\leq\|\nabla
e^{(\Delta-\frac{1}{d})\tilde{t}}v_{0}\|_{L^{\infty}}+\frac{\xi_{2}}{d}\int_{0}^{\tilde{t}}\|\nabla
e^{(\Delta-\frac{1}{d})(\tilde{t}-s)}(\nabla w\cdot\nabla
v)(\cdot,s)\|_{L^{\infty}}ds$ $\displaystyle\ \ \ \
+\frac{\xi_{2}}{d}\int_{0}^{\tilde{t}}\|\nabla
e^{(\Delta-\frac{1}{d})(\tilde{t}-s)}v(\cdot,s)(\bar{u}(s)-u(\cdot,s))\|_{L^{\infty}}ds$
$\displaystyle\ \ \ \ +\frac{1}{d}\int_{0}^{\tilde{t}}\|\nabla
e^{(\Delta-\frac{1}{d})(\tilde{t}-s)}u(\cdot,s)\|_{L^{\infty}}ds$
$\displaystyle\leq c_{40}\|\nabla
v_{0}\|_{L^{\infty}}+\frac{c_{40}\xi_{2}}{d}\int_{0}^{\tilde{t}}(1+(\tilde{t}-s)^{-\frac{2}{3}})e^{-\lambda_{1}(\tilde{t}-s)}\|\nabla
v\|_{L^{6n}}\|\nabla w\|_{L^{6n}}ds$ $\displaystyle\ \ \ \
+\frac{c_{40}\xi_{2}}{d}\int_{0}^{\tilde{t}}(1+(\tilde{t}-s)^{-\frac{2}{3}})e^{-\lambda_{1}(\tilde{t}-s)}\|v\|_{L^{\infty}}\|\bar{u}-u\|_{L^{3n}}ds$
$\displaystyle\ \ \ \
+\frac{c_{40}}{d}\int_{0}^{\tilde{t}}(1+(\tilde{t}-s)^{-\frac{2}{3}})e^{-\lambda_{1}(\tilde{t}-s)}\|u\|_{L^{3n}}ds$
$\displaystyle\leq
c_{41}\Bigl{(}1+\frac{\xi_{2}M_{2}^{\frac{1}{2n}}(3n)}{d\left((1+M_{0}\xi_{2})^{-3n}d^{-1}\chi^{2}M_{0}^{2-3n}\right)^{\frac{1}{6n}}}+\frac{\xi_{2}}{d}M_{0}M_{2}^{\frac{1}{3n}}(3n)+\frac{M_{2}^{\frac{1}{3n}}(3n)}{d}\Bigr{)},$

which is our desired qualitative bound in (3.77) since
$\tilde{t}\in(0,dT_{\text{max}})$ was arbitrary. ∎

###### Proof of Theorem 1.1.

Based on the extensibility criterion (2.1) in Lemma 2.1, the qualitative
boundedness described in Theorem 1.1 can be traced out from Lemma 3.1, Lemma
3.6 and Lemma 3.7. The qualitative bound for $\|v\|_{L^{\infty}}$ in (1.7) can
be easily seen from Lemma 3.1. The bounds for $\|\nabla w\|_{L^{\infty}}$ in
(1.8) comes mainly from Lemma 3.5 with $k=n+1$. ∎

##  4\. Higher order regularity of solutions

In section 3, we have established the qualitative boundedness and thus global
existence of solutions. To study large time behavior of global bounded
solutions, we need further to enhance regularity properties of bounded
solutions.

######  Lemma 4.1.

Let $(u,v,w)$ be the global and bounded classical solution of (1.4) obtained
in Theorem 1.1. Then there exist $\sigma\in(0,1)$ and $C_{7}>0$ such that

$\|u(\cdot,t)\|_{C^{\sigma,\frac{\sigma}{2}}(\bar{\Omega}\times[t,t+1])}\leq
C_{7},\quad\quad\forall t\geq 1.$ (4.1)

###### Proof.

In light of Theorem 1.1, the global classical solution $(u,v,w)$ satisfies

$u>0,\ v>0,\ u+v+|\nabla v|+|\nabla w|\leq M\text{ on
}\Omega\times(0,\infty).$ (4.2)

We now rewrite the first equation of (1.4) in the following form:

$u_{t}=\nabla\cdot D(x,t,\nabla
u)+R(x,t),\quad\quad(x,t)\in\Omega\times(0,\infty),$

where $D(x,t,\eta)=\eta-\chi(u\nabla v)(x,t)+\xi_{1}(u\nabla w)(x,t)$ and
$R(x,t)=(au-\mu u^{\theta+1})(x,t)$.

In view of the boundedness in (4.2) and the Young inequality, we readily
deduce

$\begin{cases}D(x,t,\eta)\cdot\eta\geq\frac{1}{2}|\eta|^{2}-\frac{(\chi+\xi_{1})^{2}M^{4}}{2},\
\ |D(x,t,\eta)|\leq|\eta|+(\chi+\xi_{1})M^{2},\\\\[7.11317pt] |R(x,t)|\leq
aM+\mu M^{\theta+1},\ \ \
\forall(x,t,\eta)\in\Omega\times(0,\infty)\times\mathbb{R}^{n}.\end{cases}$

Then (4.1) follows from the Hölder regularity for parabolic equations [34,
Theorem 1.3]. ∎

######  Lemma 4.2.

There exists a constant $C_{8}>0$ such that the global bounded solution of
(1.4) obtained in Theorem 1.1 fulfills

$\|\nabla u(\cdot,t)\|_{L^{2n}}\leq C_{8},\quad\quad\forall t\geq 1.$ (4.3)

###### Proof.

Inspired from [23, Section 3.3], we first establish an $L^{2}$-bound for
$\Delta v$. To this end, we first recall from (4.2) and (3.14) with $p=2$ of
Lemma 3.2 that

$\frac{d}{dt}\int_{\Omega}u^{2}+\int_{\Omega}u^{2}+\int_{\Omega}|\nabla
u|^{2}\leq c_{42}.$ (4.4)

Next, we take $\nabla$ to the $v$-equation, and then take dot product with
$-\nabla\Delta v$, and finally use the $w$-equation and repeatedly use
integration by parts to derive from (1.4) that

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\Delta
v|^{2}+\int_{\Omega}|\Delta v|^{2}+d\int_{\Omega}|\nabla\Delta v|^{2}\\\
&=-\xi_{2}\int_{\Omega}\nabla\left(\nabla v\cdot\nabla
w+\bar{u}v-uv\right)\cdot\nabla\Delta v-\int_{\Omega}\nabla u\cdot\nabla\Delta
v\\\ &=-\xi_{2}\int_{\Omega}\left(D^{2}v\cdot\nabla w+D^{2}w\cdot\nabla
v\right)\cdot\nabla\Delta v+\xi_{2}\int_{\Omega}\left(u-\bar{u}\right)\nabla
v\cdot\nabla\Delta v\\\ &\;\;\;\;\;+\int_{\Omega}\left(\xi_{2}v-1\right)\nabla
u\cdot\nabla\Delta v.\end{split}$ (4.5)

Based on (4.4) and (4.5), by the boundedness (4.2), the elliptic estimate
$\|D^{2}w\|_{L^{2}}\leq c_{43}\|u\|_{L^{2}}\leq c_{44}$, the Gagliardo-
Nirenberg interpolation inequality and the $H^{3}$-elliptic estimate (cf. [43,
(4.19)]), we obtain that

$\begin{split}&-\xi_{2}\int_{\Omega}\left(D^{2}v\cdot\nabla
w+D^{2}w\cdot\nabla v\right)\nabla\Delta v\\\ &\leq
M\xi_{2}\|D^{2}v\|_{L^{2}}\|\nabla\Delta v\|_{L^{2}}+\frac{d}{7}\|\nabla\Delta
v\|_{L^{2}}^{2}+c_{45}\\\ &\leq
c_{46}\left(\|D^{3}v\|_{L^{2}}^{\frac{2}{3}}\|v\|_{L^{2}}^{\frac{1}{3}}+\|v\|_{L^{2}}\right)\|\nabla\Delta
v\|_{L^{2}}+\frac{d}{7}\|\nabla\Delta v\|_{L^{2}}^{2}+c_{45}\\\ &\leq
c_{47}\|v\|_{H^{3}}^{\frac{2}{3}}\|\nabla\Delta
v\|_{L^{2}}+\frac{d}{6}\|\nabla\Delta v\|_{L^{2}}^{2}+c_{47}\\\ &\leq
c_{48}\left(\|\Delta
v\|_{H^{1}}+\|v\|_{L^{2}}\right)^{\frac{2}{3}}\|\nabla\Delta
v\|_{L^{2}}+\frac{d}{6}\|\nabla\Delta v\|_{L^{2}}^{2}+c_{47}\\\ &\leq
c_{49}\left(\|\nabla\Delta
v\|_{L^{2}}+\|v\|_{L^{2}}\right)^{\frac{2}{3}}\|\nabla\Delta
v\|_{L^{2}}+\frac{d}{6}\|\nabla\Delta v\|_{L^{2}}^{2}+c_{47}\\\
&\leq\frac{d}{5}\|\nabla\Delta v\|_{L^{2}}^{2}+c_{50}.\end{split}$ (4.6)

In easier ways, we again use the boundedness (4.2) to estimate

$\begin{split}&\xi_{2}\int_{\Omega}\left(u-\bar{u}\right)\nabla
v\cdot\nabla\Delta v+\int_{\Omega}\left(\xi_{2}v-1\right)\nabla
u\cdot\nabla\Delta v\\\ &\leq\frac{3d}{5}\int_{\Omega}|\nabla\Delta
v|^{2}+\frac{(1+M\xi_{2})^{2}}{2d}\int_{\Omega}|\nabla
u|^{2}+c_{51}.\end{split}$ (4.7)

Substituting (4.6) and (4.7) into (4.5), we conclude that

$\frac{d}{dt}\int_{\Omega}|\Delta v|^{2}+2\int_{\Omega}|\Delta
v|^{2}\leq\frac{(1+M\xi_{2})^{2}}{d}\int_{\Omega}|\nabla
u|^{2}+2c_{50}+2c_{51}.$ (4.8)

An obvious combination from (4.4) and (4.8) enables one to derive that

$\frac{d}{dt}\int_{\Omega}\left(\frac{(1+M\xi_{2})^{2}}{d}u^{2}+|\Delta
v|^{2}\right)+\int_{\Omega}\left(\frac{(1+M\xi_{2})^{2}}{d}u^{2}+|\Delta
v|^{2}\right)\leq c_{52},$

which along with $H^{2}$-elliptic estimate yields a uniform $H^{2}$-bound for
$v$:

$\|D^{2}v(\cdot,t)\|_{L^{2}}+\|\Delta v(\cdot,t)\|_{L^{2}}\leq
c_{53},\quad\quad\forall t\geq 1.$ (4.9)

Next, we proceed to derive an $L^{2}$-bound for $\nabla u$. For this, we
compute from (1.4) that

$\begin{split}&\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}+\int_{\Omega}|\nabla
u|^{2}+2\int_{\Omega}|\Delta u|^{2}\\\ &=2\chi\int_{\Omega}\left(\nabla
u\cdot\nabla v+u\Delta v\right)\Delta u-2\xi_{1}\int_{\Omega}\left(\nabla
u\cdot\nabla w+u\bar{u}-u^{2}\right)\Delta u\\\ &\ \ -2\int_{\Omega}(au-\mu
u^{\theta+1})\Delta u+\int_{\Omega}|\nabla u|^{2}.\end{split}$ (4.10)

Using the boundedness (4.2), (4.9), the Gagliardo-Nirenberg inequality and the
$H^{2}$-elliptic estimate, we deduce that

$\begin{split}&2\chi\int_{\Omega}\left(\nabla u\nabla v+u\Delta v\right)\Delta
u-2\xi_{1}\int_{\Omega}\left(\nabla u\nabla w+u\bar{u}-u^{2}\right)\Delta u\\\
&\ \ -2\int_{\Omega}(au-\mu u^{\theta+1})\Delta u+\int_{\Omega}|\nabla
u|^{2}\\\ &\leq\int_{\Omega}|\Delta
u|^{2}+\left[1+4M^{2}(\chi^{2}+\xi^{2}_{1})\right]\int_{\Omega}|\nabla
u|^{2}+4M^{2}\chi^{2}\int_{\Omega}|\Delta v|^{2}+c_{54}\\\
&\leq\int_{\Omega}|\Delta
u|^{2}+c_{55}\left(\|D^{2}u\|_{L^{2}}^{\frac{1}{2}}\|u\|_{L^{2}}^{\frac{1}{2}}+\|u\|_{L^{2}}\right)^{2}+c_{55}\\\
&\leq\int_{\Omega}|\Delta u|^{2}+c_{56}\|D^{2}u\|_{L^{2}}+c_{56}\\\
&\leq\int_{\Omega}|\Delta u|^{2}+c_{57}\left(\|\Delta
u\|_{L^{2}}+\|u\|_{L^{2}}\right)+c_{57}\\\ &\leq 2\int_{\Omega}|\Delta
u|^{2}+c_{58}.\end{split}$

Substituting this into (4.10) entails a simple ODI as follows:

$\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}+\int_{\Omega}|\nabla u|^{2}\leq
c_{58},$

yielding immediately a uniform $L^{2}$-bound for $\nabla u$:

$\|\nabla u(\cdot,t)\|_{L^{2}}\leq c_{59},\quad\quad\forall t\geq 1.$ (4.11)

We are now ready to show the uniform boundedness of $\|\nabla
u(\cdot,t)\|_{L^{2n}}$. Again, we apply integration by parts to compute from
(1.4) that

$\begin{split}\frac{1}{2n}\frac{d}{dt}\int_{\Omega}|\nabla
u|^{2n}&=\int_{\Omega}|\nabla u|^{2n-2}\nabla u\cdot\nabla\Delta
u-\chi\int_{\Omega}|\nabla u|^{2n-2}\nabla u\cdot\nabla(\nabla\cdot(u\nabla
v))\\\ &\;\;\;+\xi_{1}\int_{\Omega}|\nabla u|^{2n-2}\nabla
u\cdot\nabla(\nabla\cdot(u\nabla w))\\\ &\;\;\;+\int_{\Omega}|\nabla
u|^{2n-2}\nabla u\cdot\nabla(au-\mu
u^{\theta+1})=:\sum_{i=1}^{4}J_{i}.\end{split}$ (4.12)

For $J_{1}$, noticing the fact that $2\nabla u\cdot\nabla\Delta
u=\Delta|\nabla u|^{2}-2|D^{2}u|^{2}$ and the $L^{2}$-boundedness of $\nabla
u$ in (4.11), we employ the way to control boundary integral as done in (3.36)
and the GN inequality to estimate that

$\begin{split}J_{1}&=\frac{1}{2}\int_{\Omega}|\nabla u|^{2n-2}\Delta|\nabla
u|^{2}-\int_{\Omega}|\nabla u|^{2n-2}|D^{2}u|^{2}\\\
&=\frac{1}{2}\int_{\partial\Omega}|\nabla u|^{2n-2}\frac{\partial|\nabla
u|^{2}}{\partial\nu}-\frac{(n-1)}{2}\int_{\Omega}|\nabla
u|^{2n-4}|\nabla|\nabla u|^{2}|^{2}-\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|^{2}\\\ &\leq-\frac{(n-1)}{3}\int_{\Omega}|\nabla
u|^{2n-4}|\nabla|\nabla u|^{2}|^{2}-\frac{1}{2}\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|^{2}+c_{60}.\end{split}$ (4.13)

By the inequality $|\Delta u|\leq\sqrt{n}|D^{2}u|$ and the boundedness (4.2)
and (4.9), we use integration by parts to bound $J_{2}$ as follows:

$\begin{split}J_{2}=&\chi\int_{\Omega}\nabla|\nabla u|^{2n-2}\cdot\nabla
u\nabla\cdot(u\nabla v)+\chi\int_{\Omega}|\nabla u|^{2n-2}\Delta
u\nabla\cdot(u\nabla v)\\\ =&\chi(n-1)\int_{\Omega}|\nabla
u|^{2(n-2)}\nabla|\nabla u|^{2}\cdot\nabla u(\nabla u\cdot\nabla v+u\Delta
v)\\\ &+\chi\int_{\Omega}|\nabla u|^{2n-2}\Delta u(\nabla u\cdot\nabla
v+u\Delta v)\\\ \leq&M\chi(n-1)\int_{\Omega}|\nabla u|^{2n-3}|\nabla|\nabla
u|^{2}|(|\nabla u|+|\Delta v|)\\\ &\;\;\;+M\chi\sqrt{n}\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|(|\nabla u|+|\Delta v|)\\\
\leq&\frac{(n-1)}{8}\int_{\Omega}|\nabla u|^{2n-4}|\nabla|\nabla
u|^{2}|^{2}+\frac{1}{8}\int_{\Omega}|\nabla u|^{2n-2}|D^{2}u|^{2}\\\
&+4(2n-1)M^{2}\chi^{2}\int_{\Omega}|\nabla
u|^{2n}+4(2n-1)M^{2}\chi^{2}\int_{\Omega}|\nabla u|^{2n-2}|\Delta
v|^{2}.\end{split}$ (4.14)

In a similar way, since $|\Delta w|=|u-\bar{u}|\leq 2M$, and using Young’s
inequality, we estimate $J_{3}$ as follows:

$\begin{split}J_{3}=&-\xi_{1}\int_{\Omega}\nabla|\nabla u|^{2n-2}\cdot\nabla
u\nabla\cdot(u\nabla w)-\xi_{1}\int_{\Omega}|\nabla u|^{2n-2}\Delta
u\cdot\nabla(u\nabla w)\\\ \leq&M\xi_{1}(n-1)\int_{\Omega}|\nabla
u|^{2n-3}|\nabla|\nabla u|^{2}|(|\nabla u|+|\Delta w|)\\\
&\;\;\;+M\xi_{1}\sqrt{n}\int_{\Omega}|\nabla u|^{2n-2}|D^{2}u|(|\nabla
u|+|\Delta w|)\\\ \leq&\frac{(n-1)}{8}\int_{\Omega}|\nabla
u|^{2n-4}|\nabla|\nabla u|^{2}|^{2}+\frac{1}{8}\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|^{2}\\\ &+4(2n-1)M^{2}\xi_{1}^{2}\int_{\Omega}|\nabla
u|^{2n}+16(2n-1)M^{4}\xi_{1}^{2}\int_{\Omega}|\nabla u|^{2n-2}\\\
\leq&\frac{(n-1)}{8}\int_{\Omega}|\nabla u|^{2n-4}|\nabla|\nabla
u|^{2}|^{2}+\frac{1}{8}\int_{\Omega}|\nabla u|^{2n-2}|D^{2}u|^{2}\\\
&+2(4n-1)M^{2}\xi_{1}^{2}\int_{\Omega}|\nabla u|^{2n}+c_{61}.\end{split}$
(4.15)

At last, the term $J_{4}$ can be easily computed:

$J_{4}=a\int_{\Omega}|\nabla
u|^{2n}-\mu(\theta+1)\int_{\Omega}u^{\theta}|\nabla u|^{2n}.$ (4.16)

Substituting (4.13), (4.14), (4.15) and (4.16) into (4.12) and using Young’s
inequality, one has a differential inequality of the form:

$\begin{split}&\frac{d}{dt}\int_{\Omega}|\nabla u|^{2n}+2n\int_{\Omega}|\nabla
u|^{2n}+\frac{(n-1)n}{6}\int_{\Omega}|\nabla u|^{2n-4}|\nabla|\nabla
u|^{2}|^{2}\\\ &\ \ \ +\frac{n}{2}\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|^{2}\\\ &\leq c_{62}\int_{\Omega}|\nabla
u|^{2n}+c_{62}\int_{\Omega}|\nabla u|^{2n-2}|\Delta v|^{2}+c_{62}\\\ &\leq
c_{63}\int_{\Omega}|\nabla u|^{2n}+c_{63}\int_{\Omega}|\Delta
v|^{2n}+c_{63}.\end{split}$ (4.17)

Since $\|u(\cdot,t)\|_{L^{\infty}}\leq M$ in (4.2), the Young’s inequality
enables one to deduce that

$\displaystyle 2c_{63}\int_{\Omega}|\nabla u|^{2n}$
$\displaystyle=2c_{63}\left(-(n-1)\int_{\Omega}u|\nabla u|^{2n-4}\nabla|\nabla
u|^{2}\cdot\nabla u-\int_{\Omega}u|\nabla u|^{2n-2}\Delta u\right)$
$\displaystyle\leq 2c_{63}M\left((n-1)\int_{\Omega}|\nabla
u|^{2n-3}|\nabla|\nabla u|^{2}|+\sqrt{n}\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|\right)$
$\displaystyle\leq\frac{n(n-1)}{6}\int_{\Omega}|\nabla u|^{2n-4}|\nabla|\nabla
u|^{2}|^{2}+\frac{n}{2}\int_{\Omega}|\nabla
u|^{2n-2}|D^{2}u|^{2}+c_{64}\int_{\Omega}|\nabla u|^{2n-2}$
$\displaystyle\leq\frac{n(n-1)}{6}\int_{\Omega}|\nabla u|^{2n-4}|\nabla|\nabla
u|^{2}|^{2}+\frac{n}{2}\int_{\Omega}|\nabla u|^{2n-2}|D^{2}u|^{2}$
$\displaystyle\;\;\;+c_{63}\int_{\Omega}|\nabla u|^{2n}+c_{65},$

which, upon being substituted into (4.17), allows us to conclude

$\displaystyle\frac{d}{dt}\int_{\Omega}|\nabla u|^{2n}+2n\int_{\Omega}|\nabla
u|^{2n}\leq c_{63}\int_{\Omega}|\Delta v|^{2n}+c_{66}.$

Solving this simple differential inequality via integrating factor method, we
get

$\int_{\Omega}|\nabla
u(\cdot,t)|^{2n}\leq\frac{c_{66}}{2n}+\int_{\Omega}|\nabla
u(\cdot,1)|^{2n}+c_{63}e^{-2nt}\int_{1}^{t}e^{2ns}\int_{\Omega}|\Delta
v(\cdot,s)|^{2n}ds,\quad\forall t\geq 1.$ (4.18)

Next, we apply the widely known maximal Sobolev regularity to the $v$-equation
to bound the third term on the right. To this end, with the help of the
$w$-equation, we first rewrite the $v$-equation in (1.4) in the following
form:

$\begin{split}v_{t}&=d\Delta v-v+\xi_{2}\nabla v\cdot\nabla
w+\xi_{2}v(\bar{u}-u)+u.\\\ \end{split}$ (4.19)

Let $h(x,t):=\xi_{2}\nabla v\cdot\nabla w+\xi_{2}v(\bar{u}-u)+u$. Then it
follows from the boundedness (4.2) that

$|h|\leq M^{2}\xi_{2}+2M^{2}\xi_{2}+M=3M^{2}\xi_{2}+M\text{ on
}\Omega\times(0,\infty).$ (4.20)

Now, an application of the maximal Sobolev regularity to (4.19) with (4.20)
shows that

$\begin{split}\int_{1}^{t}e^{2ns}\int_{\Omega}|\Delta v(\cdot,t)|^{2n}&\leq
c_{67}\left(e^{2n}\int_{\Omega}|\Delta
v(\cdot,1)|^{2n}+\int_{1}^{t}e^{2ns}\int_{\Omega}|h|^{2n}\right)\\\ &\leq
c_{67}\left(e^{2n}\int_{\Omega}|\Delta
v(\cdot,1)|^{2n}+\frac{(3M^{2}\xi_{2}+M)^{2n}|\Omega|}{2n}e^{2nt}\right),\end{split}$

which, substituted into (4.18), gives, for $\forall t\geq 1$,

$\int_{\Omega}|\nabla
u(\cdot,t)|^{2n}\leq\frac{c_{66}}{2n}+\int_{\Omega}|\nabla
u(\cdot,1)|^{2n}+c_{63}c_{67}\left(\int_{\Omega}|\Delta
v(\cdot,1)|^{2n}+\frac{(3M^{2}\xi_{2}+M)^{2n}|\Omega|}{2n}\right),$

yielding our desired gradient estimate (4.3). ∎

##  5\. Global stability: Proof of Theorem 1.3

Given the enhanced regularity properties in the preceding section, we shall
examine the long time dynamics of bounded solutions as obtained in Theorem
1.1. Under certain conditions, we shall show convergence and exponential
convergence of bounded solutions to the unique constant steady state as time
goes to infinity.

###  5.1. Case 1: $a=\mu=0$

In this case, the system (1.4) becomes

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+\xi_{1}\nabla\cdot(u\nabla w),&x\in\Omega,t>0,\\\ v_{t}=d\Delta
v+\xi_{2}\nabla\cdot(v\nabla w)+u-v,&x\in\Omega,t>0,\\\ 0=\Delta
w+u-\bar{u}_{0},\ \ \int_{\Omega}w=0,&x\in\Omega,t>0,\\\ \frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial
w}{\partial\nu}=0,&x\in\partial\Omega,t>0,\\\
u(x,0)=u_{0}(x),~{}v(x,0)=v_{0}(x),&x\in\Omega.\end{cases}$ (5.1)

######  Lemma 5.1.

The global classical solution of (5.1) satisfies the following identity:

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)+\int_{\Omega}\frac{|\nabla u|^{2}}{u}+d\chi\int_{\Omega}|\Delta
v|^{2}+\chi\int_{\Omega}|\nabla v|^{2}+\xi_{1}\int_{\Omega}|\Delta w|^{2}\\\
&=2\chi\int_{\Omega}\Delta w\cdot\Delta
v-\chi\xi_{2}\int_{\Omega}\nabla\cdot(v\nabla w)\Delta v.\end{split}$ (5.2)

###### Proof.

Multiplying the first equation of (5.1) by $\ln u-\ln\bar{u}$, we have

$\frac{d}{dt}\int_{\Omega}u\ln\frac{u}{\bar{u}}+\int_{\Omega}\frac{|\nabla
u|^{2}}{u}=\chi\int_{\Omega}\nabla u\cdot\nabla v-\xi_{1}\int_{\Omega}\nabla
u\cdot\nabla w.$ (5.3)

We multiply the second equation of (5.1) by $-\Delta v$ to obtain

$\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla v|^{2}+d\int_{\Omega}|\Delta
v|^{2}+\int_{\Omega}|\nabla v|^{2}=\int_{\Omega}\nabla u\cdot\nabla
v-\xi_{2}\int_{\Omega}\nabla\cdot(v\nabla w)\Delta v.$ (5.4)

Then multiplying (5.4) by $\chi$ and adding it to (5.3), and using the facts
$u=\bar{u}_{0}-\Delta w$ and $\int_{\Omega}\Delta v=0=\int_{\Omega}\Delta w$
due to homogeneous boundary conditions, we obtain

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)+\int_{\Omega}\frac{|\nabla u|^{2}}{u}+d\chi\int_{\Omega}|\Delta
v|^{2}+\chi\int_{\Omega}|\nabla v|^{2}\\\ &=2\chi\int_{\Omega}\nabla
u\cdot\nabla v-\xi_{1}\int_{\Omega}\nabla u\cdot\nabla
w-\chi\xi_{2}\int_{\Omega}\nabla\cdot(v\nabla w)\Delta v\\\
&=2\chi\int_{\Omega}\Delta w\cdot\Delta v-\xi_{1}\int_{\Omega}|\Delta
w|^{2}-\chi\xi_{2}\int_{\Omega}\nabla\cdot(v\nabla w)\Delta v,\end{split}$

which gives rise to (5.2). ∎

In the limiting case of $\chi=0$, there is no bad terms to be estimated by
(5.2), we can directly jump to (5.15) below, and hence its proof is simpler.
We proceed to treat the hard case of $\chi>0$, for that purpose, we denote

$A=A(d)=\sup_{(x,t)\in\Omega\times(0,\infty)}v(x,t),\quad\quad\quad
B=B(d)=\sup_{(x,t)\in\Omega\times(0,\infty)}|\nabla w(x,t)|.$ (5.5)

Then by the qualitative bounds for $v$ and $\nabla w$ in (1.7) and (1.8) with
(1.9) and (1.10), we see that they are bounded for large $d$ and hence

$d_{0}(\chi)=\inf\left\\{\hat{d}_{0}>0:\
\left(d-\frac{\left(2+A\xi_{2}\right)^{2}\chi}{4\xi_{1}}-\frac{B^{2}\xi_{2}^{2}}{4}\right)\chi\geq
0\text{ for all }d\geq\hat{d}_{0}\right\\}<+\infty.$ (5.6)

This allows us to include the the limiting case of $\chi=0$, which corresponds
to $d_{0}(0)=0$.

######  Lemma 5.2.

Assume $\xi_{1}\geq\xi_{0}\chi^{2}$ and $d\geq d_{0}(\chi)$. Then the global
classical solution $(u,v,w)$ of (5.1) satisfies the following estimates:

$\int_{1}^{t}\int_{\Omega}\frac{|\nabla u|^{2}}{u}\leq C_{9},\quad\quad\forall
t>1.$ (5.7)

In addition, if $d>d_{0}(\chi)$, then there exists $\zeta_{1}>0$ such that

$\|u(\cdot,t)-\bar{u}_{0}\|_{L^{1}}+\|\nabla v(\cdot,t)\|_{L^{2}}\leq
C_{10}e^{-\zeta_{1}t},\quad\quad\forall t>0.$ (5.8)

###### Proof.

Using the definitions of $A$ and $B$ in (5.5), we infer from (5.2) of Lemma
5.1 that

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)+\int_{\Omega}\frac{|\nabla u|^{2}}{u}+d\chi\int_{\Omega}|\Delta
v|^{2}+\chi\int_{\Omega}|\nabla v|^{2}+\xi_{1}\int_{\Omega}|\Delta w|^{2}\\\
&=2\chi\int_{\Omega}\Delta w\cdot\Delta v-\chi\xi_{2}\int_{\Omega}v\Delta
w\cdot\Delta v-\xi_{2}\chi\int_{\Omega}\nabla v\cdot\nabla w\cdot\Delta v\\\
&\leq\left(2+A\xi_{2}\right)\chi\int_{\Omega}|\Delta w||\Delta
v|+B\chi\xi_{2}\int_{\Omega}|\nabla v||\Delta v|.\end{split}$ (5.9)

Using the Young’s inequality, we readily derive

$\left(2+A\xi_{2}\right)\chi\int_{\Omega}|\Delta w||\Delta
v|\leq\xi_{1}\int_{\Omega}|\Delta
w|^{2}+\frac{(2+A\xi_{2})^{2}\chi^{2}}{4\xi_{1}}\int_{\Omega}|\Delta v|^{2},$
(5.10)

and

$B\chi\xi_{2}\int_{\Omega}|\nabla v||\Delta
v|\leq(1-\varepsilon_{1})\chi\int_{\Omega}|\nabla
v|^{2}+\frac{B^{2}\chi\xi_{2}^{2}}{4(1-\varepsilon_{1})}\int_{\Omega}|\Delta
v|^{2},$ (5.11)

where, due to $d\geq d_{0}(\chi)$ with $d_{0}(\chi)$ defined by (5.6),
$\varepsilon_{1}\in[0,1)$ is defined by

$\varepsilon_{1}=\begin{cases}0,&\mathrm{if}\ \ \ d=d_{0}(\chi),\\\
\frac{1}{2}\left(d-\frac{\left(2+A\xi_{2}\right)^{2}\chi}{4\xi_{1}}-\frac{B^{2}\xi_{2}^{2}}{4}\right)\left(d-\frac{\left(2+A\xi_{2}\right)^{2}\chi}{4\xi_{1}}\right)^{-1},&\mathrm{if}\
\ \ d>d_{0}(\chi).\\\ \end{cases}$ (5.12)

A substitution of (5.10) and (5.11) into (5.9) gives

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)+\int_{\Omega}\frac{|\nabla
u|^{2}}{u}+\varepsilon_{1}\chi\int_{\Omega}|\nabla v|^{2}\\\\[7.11317pt]
&+\left(d-\frac{(2+A\xi_{2})^{2}\chi}{4\xi_{1}}-\frac{B^{2}\xi_{2}^{2}}{4(1-\varepsilon_{1})}\right)\chi\int_{\Omega}|\Delta
v|^{2}\leq 0.\end{split}$ (5.13)

Since $d\geq d_{0}(\chi)$, using the definitions of $d_{0}$ and
$\varepsilon_{1}$ in (5.6) and (5.12), one has

$\left(d-\frac{(2+A\xi_{2})^{2}\chi}{4\xi_{1}}-\frac{B^{2}\xi_{2}^{2}}{4(1-\varepsilon_{1})}\right)\chi\geq
0,$

which, substituted into (5.13), yields

$\frac{d}{dt}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)+\int_{\Omega}\frac{|\nabla
u|^{2}}{u}+\varepsilon_{1}\chi\int_{\Omega}|\nabla v|^{2}\leq 0.$ (5.14)

Then an integration with respect to $t$ over $(1,t)$ shows that

$\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}+\int_{1}^{t}\int_{\Omega}\frac{|\nabla
u|^{2}}{u}\leq\int_{\Omega}u(\cdot,1)\ln\frac{u(\cdot,1)}{\bar{u}_{0}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v(\cdot,1)|^{2}.$ (5.15)

A use of Lemma 2.6 along with the fact $\bar{u}=\bar{u}_{0}$ entails

$\int_{\Omega}u\ln\frac{u}{\bar{u}}\geq\frac{1}{2\bar{u}_{0}}\|u-\bar{u}_{0}\|_{L^{1}}^{2}\geq
0,$ (5.16)

and

$\int_{\Omega}u(\cdot,1)\ln\frac{u(\cdot,1)}{\bar{u}_{0}}\leq\frac{1}{\bar{u}_{0}}\|u(\cdot,1)-\bar{u}_{0}\|_{L^{2}}^{2}.$
(5.17)

Substituting (5.16) and (5.17) into (5.15), we obtain (5.7) directly.

On the other hand, using Lemma 2.6 again and noting the fact of
$\|u\|_{L^{\infty}}\leq M$, we use the Poincaré inequality to derive that

$\int_{\Omega}u\ln\frac{u}{\bar{u}}\leq\frac{1}{\bar{u}}\|u-\bar{u}\|_{L^{2}}^{2}\leq\frac{c_{68}}{\bar{u}_{0}}\|\nabla
u\|_{L^{2}}^{2}\leq\frac{c_{68}M}{\bar{u}_{0}}\int_{\Omega}\frac{|\nabla
u|^{2}}{u},$

which yields

$\frac{\bar{u}_{0}}{c_{68}M}\int_{\Omega}u\ln\frac{u}{\bar{u}}\leq\int_{\Omega}\frac{|\nabla
u|^{2}}{u}.$ (5.18)

Substituting (5.18) into (5.14), and recalling the fact $\varepsilon_{1}>0$
due to $d>d_{0}(\chi)$, then we find a positive constant
$c_{69}:=\min\\{\frac{\bar{u}_{0}}{c_{68}M},2\varepsilon_{1}\\}$ such that

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)+c_{69}\left(\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\right)\leq 0,\end{split}$

which immediately entails, for $t\geq 1$,

$\int_{\Omega}u\ln\frac{u}{\bar{u}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v|^{2}\leq\left(\int_{\Omega}u(\cdot,1)\ln\frac{u(\cdot,1)}{\bar{u}_{0}}+\frac{\chi}{2}\int_{\Omega}|\nabla
v(\cdot,1)|^{2}\right)e^{-c_{69}(t-1)}.$ (5.19)

Then the combination of (5.16), (5.17) and (5.19) implies (5.8). ∎

######  Lemma 5.3.

Under Lemma 5.2, the $u,w$-components of the global bounded classical solution
of (5.1) fulfills the following properties:

  * (uc1)

If $d\geq d_{0}(\chi)$, then $(u,w)$ decays to $(\bar{u}_{0},0)$ uniformly:

$\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\to 0\ \
\mathrm{as}\ \ t\to\infty.$ (5.20)

  * (uc2)

If $d>d_{0}(\chi)$, then $(u,w)$ decays exponentially to $(\bar{u}_{0},0)$:
for some $\zeta_{2}>0$,

$\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\leq
C_{11}e^{-\zeta_{2}t},\ \ \ \ \forall t>0.$ (5.21)

###### Proof.

Notice from Theorem 1.1 that $\|u(\cdot,t)\|_{L^{\infty}}$ is uniformly
bounded, and thus the space-time estimate (5.7) along with Poincaré inequality
ensures

$\int_{1}^{\infty}\int_{\Omega}|u-\bar{u}_{0}|^{2}\leq
c_{69}\int_{1}^{\infty}\int_{\Omega}|\nabla u|^{2}\leq c_{70}.$ (5.22)

Then the uniform continuity of $\|u-\bar{u}_{0}\|_{L^{2}}^{2}$ implied by
(4.1) shows that

$\|u-\bar{u}_{0}\|_{L^{2}}\rightarrow 0\text{ as }t\rightarrow\infty.$ (5.23)

Thus, with the boundedness of $\|\nabla u\|_{L^{2n}}$ in (4.3), the Gagliardo-
Nirenberg inequality gives

$\begin{split}\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}\leq&c_{71}\left(\|\nabla
u(\cdot,t)\|_{L^{2n}}^{\frac{n}{n+1}}\|u(\cdot,t)-\bar{u}_{0}\|_{L^{2}}^{\frac{1}{n+1}}+\|u(\cdot,t)-\bar{u}_{0}\|_{L^{2}}\right)\\\
\leq&c_{72}\left(\|u(\cdot,t)-\bar{u}_{0}\|_{L^{2}}^{\frac{1}{n+1}}+\|u(\cdot,t)-\bar{u}_{0}\|_{L^{2}}\right)\rightarrow
0\text{ as }t\rightarrow\infty.\end{split}$ (5.24)

This establishes the $u$-convergence in (5.20). We here also provide an
alterative short proof based on [37, Lemma 3.10]. Indeed, assume to the
contrary, then there would exist some sequences
$(x_{j})_{j\in\mathbb{N}}\subset\Omega$ and
$(t_{j})_{j\in\mathbb{N}}\subset(0,\infty)$ satisfying $t_{j}\to\infty$ as
$j\to\infty$ such that

$|u(x_{j},t_{j})-\bar{u}_{0}|\geq c_{73},\ \ \ \forall j\in\mathbb{N}.$

The uniform continuity of $u$ due to Lemma 4.1 warrants there exist $r>0$ and
$\delta>0$ such that, for any $j\in\mathbb{N}$,

$|u-\bar{u}_{0}|\geq\frac{c_{73}}{2}\text{ on
}B_{r}(x_{j})\cap\Omega\times(t_{j},t_{j}+\delta).$ (5.25)

The smoothness of $\partial\Omega$ shows that

$|B_{r}(x_{j})\cap\Omega|\geq c_{74},\ \ \forall x_{j}\in\Omega.$ (5.26)

Therefore, for all $j\in\mathbb{N}$, it follows from (5.25) and (5.26) that

$\int_{t_{j}}^{t_{j}+\delta}\int_{\Omega}|u-\bar{u}_{0}|^{2}\geq\int_{t_{j}}^{t_{j}+\delta}\int_{B_{r}(x_{j})\cap\Omega}|u-\bar{u}_{0}|^{2}\geq\frac{c_{73}^{2}c_{74}\delta}{4},$
(5.27)

which clearly contradicts the following fact due to (5.22):

$\int_{t_{j}}^{t_{j}+\delta}\int_{\Omega}|u-\bar{u}_{0}|^{2}\leq\int_{t_{j}}^{\infty}\int_{\Omega}|u-\bar{u}_{0}|^{2}\to
0\ \mathrm{as}\ \ j\to\infty.$

This contradiction gives rise to (5.20).

In the case of $d>d_{0}(\chi)$, we shall apply Lemma 4.2 and (5.8) to show the
$L^{\infty}$\- exponential decay (5.21). Indeed, employing the Gagliardo-
Nirenberg inequality, the $L^{1}$-exponential decay (5.8) and the boundedness
of $\|\nabla u\|_{L^{2n}}$ in Lemma 4.2, we conclude that

$\begin{split}\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}\leq&c_{75}\left(\|\nabla
u(\cdot,t)\|_{L^{2n}}^{\frac{2n}{2n+1}}\|u(\cdot,t)-\bar{u}_{0}\|_{L^{1}}^{\frac{1}{2n+1}}+\|u(\cdot,t)-\bar{u}_{0}\|_{L^{1}}\right)\\\
\leq&c_{76}e^{-\frac{\zeta_{1}}{2n+1}t},\end{split}$

which shows our $u$-exponential decay estimate (5.20).

Finally, we apply the $W^{2,\infty}$-estimate to (3.27) obtain

$\|w(\cdot,t)\|_{W^{2,\infty}}\leq
c_{77}\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}},$

hence, the assertions for the convergence for $w$ follow directly the
convergence of $u$. ∎

######  Lemma 5.4.

Under Lemma 5.2, the $v$-component of the global bounded classical solution of
(5.1) enjoys the following convergence properties:

  * (vc1)

If $d\geq d_{0}(\chi)$, then $v$ decays to $\bar{u}_{0}$ uniformly:

$\|v(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}\to 0\ \ \mathrm{as}\ \ t\to\infty.$
(5.28)

  * (vc2)

If $d>d_{0}(\chi)$, then $v$ decays exponentially to $\bar{u}_{0}$ uniformly:
for some $\zeta_{3}>0$,

$\|v(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}\leq C_{12}e^{-\zeta_{3}t},\ \ \ \
\forall t>0.$ (5.29)

###### Proof.

We first rewrite the $v$-equation in (5.1) as

$(v-\bar{u}_{0})_{t}=d\Delta(v-\bar{u}_{0})+\xi_{2}\nabla\cdot((v-\bar{u}_{0})\nabla
w)+\xi_{2}\bar{u}_{0}\Delta w+u-\bar{u}_{0}-(v-\bar{u}_{0}).$ (5.30)

Multiplying (5.30) by $v-\bar{u}_{0}$, integrating over $\Omega$ by parts and
using the fact $\Delta w=\bar{u}_{0}-u$ by (5.1), we compute that

$\begin{split}&\frac{1}{2}\frac{d}{dt}\int_{\Omega}(v-\bar{u}_{0})^{2}+d\int_{\Omega}|\nabla
v|^{2}+\int_{\Omega}(v-\bar{u}_{0})^{2}\\\
&=\frac{\xi_{2}}{2}\int_{\Omega}(v-\bar{u}_{0})^{2}\Delta
w+\xi_{2}\bar{u}_{0}\int_{\Omega}(v-\bar{u}_{0})\Delta
w+\int_{\Omega}(u-\bar{u}_{0})(v-\bar{u}_{0})\\\
&=\frac{\xi_{2}}{2}\int_{\Omega}(v-\bar{u}_{0})^{2}(\bar{u}_{0}-u)+\xi_{2}\bar{u}_{0}\int_{\Omega}(v-\bar{u}_{0})(\bar{u}_{0}-u)+\int_{\Omega}(u-\bar{u}_{0})(v-\bar{u}_{0}).\end{split}$
(5.31)

The fact $\|u-\bar{u}_{0}\|_{L^{\infty}}\to 0$ as $t\to\infty$ by (5.20) shows
there exists $t_{1}>0$ such that

$\|u(\cdot,t)-\bar{u}_{0}\|_{L^{\infty}}\leq\frac{1}{2\xi_{2}},\quad\quad
t\geq t_{1}.$ (5.32)

On the other hand, using Young’s inequality, we derive

$\xi_{2}\bar{u}_{0}\int_{\Omega}(v-\bar{u}_{0})(\bar{u}_{0}-u)\leq\frac{1}{8}\int_{\Omega}(v-\bar{u}_{0})^{2}+2\xi_{2}^{2}\bar{u}_{0}^{2}\int_{\Omega}(u-\bar{u}_{0})^{2},$
(5.33)

and

$\int_{\Omega}(u-\bar{u}_{0})(v-\bar{u}_{0})\leq\frac{1}{8}\int_{\Omega}(v-\bar{u}_{0})^{2}+2\int_{\Omega}(u-\bar{u}_{0})^{2}.$
(5.34)

Combining (5.32),(5.33), (5.34) and (5.31), we conclude, for all $t\geq
t_{1}$, that

$\frac{d}{dt}\int_{\Omega}(v-\bar{u}_{0})^{2}+2d\int_{\Omega}|\nabla
v|^{2}+\int_{\Omega}(v-\bar{u}_{0})^{2}\leq
4(1+\xi_{2}^{2}\bar{u}_{0}^{2})\int_{\Omega}(u-\bar{u}_{0})^{2},$

which, upon being solved and the fact (5.23), implies

$\begin{split}\|v(\cdot,t)-\bar{u}_{0}\|_{L^{2}}^{2}&\leq\|v(\cdot,1)-\bar{u}_{0}\|_{L^{2}}^{2}e^{-(t-1)}+4(1+\xi_{2}^{2}\bar{u}_{0}^{2})\int_{1}^{t}e^{-(t-s)}\int_{\Omega}(u(\cdot,s)-\bar{u}_{0})^{2}\\\
&\rightarrow 0\ \ \text{ as }t\rightarrow\infty.\end{split}$ (5.35)

Then, with the $L^{2}$-convergence (5.35) at hand, replacing $v$ with $u$ in
the Gagliardo-Nirenberg inequality (5.24), we readily obtain (5.28).

In the case of $d>d_{0}(\chi)$, to show $v$ decays exponentially, we first see
from (5.1) that

$\bar{v}-\bar{u}_{0}=\left(\bar{v}_{0}-\bar{u}_{0}\right)e^{-t},\ \ \ \forall
t>0.$ (5.36)

Now, we use Poincaré inequality and the exponential decays (5.8) and (5.36) to
deduce

$\begin{split}\|v(\cdot,t)-\bar{u}_{0}\|_{L^{2}}&\leq\|v-\bar{v}\|_{L^{2}}+\|\bar{v}-\bar{u}_{0}\|_{L^{2}}\\\
&\leq c_{78}\|\nabla v\|_{L^{2}}+|\bar{v}_{0}-\bar{u}_{0}||\Omega|e^{-t}\\\
&\leq c_{79}e^{-\min\\{\zeta_{1},\ 1\\}t},\ \ \forall t>0.\end{split}$

Now, since $\|\nabla v\|_{L^{2n}}$ is uniformly-in-time bounded by Theorem
1.1, replacing $v$ with $u$ in the Gagliardo-Nirenberg inequality (5.24), we
readily improve this $L^{2}$-exponential decay to $L^{\infty}$-exponential
decay of $v$ as in (5.29). ∎

###  5.2. Case 2: $a,\mu>0$

In this subsection, we shall study the large time behavior of solution for the
complete system (1.4) with $a,\mu>0$. For convenience, we rewrite it here:

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+\xi_{1}\nabla\cdot(u\nabla w)+u(a-\mu u^{\theta}),&x\in\Omega,t>0,\\\
v_{t}=d\Delta v+\xi_{2}\nabla\cdot(v\nabla w)+u-v,&x\in\Omega,t>0,\\\ 0=\Delta
w+u-\bar{u},\ \ \int_{\Omega}w=0,&x\in\Omega,t>0,\\\ \frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}=\frac{\partial
w}{\partial\nu}=0,&x\in\partial\Omega,t>0,\\\
u(x,0)=u_{0}(x),~{}v(x,0)=v_{0}(x),&x\in\Omega.\end{cases}$ (5.37)

######  Lemma 5.5.

Let $C_{p}$ denote the Poincaré constant defined by (1.13) and $(u,v,w)$ be
the solution of the system (5.37). Then

$\|\nabla w\|_{L^{2}}\leq C_{p}\|u-b\|_{L^{2}},\ \ \|\Delta
w\|_{L^{2}}\leq\|u-b\|_{L^{2}},\ \ \|w\|_{W^{2,2}}\leq C_{13}\|u-b\|_{L^{2}},\
\ b=(\frac{a}{\mu})^{\frac{1}{\theta}}.$ (5.38)

###### Proof.

By the $w$-equation in (5.1) and the Poincaré inequality due to
$\int_{\Omega}w=0$, we infer

$\int_{\Omega}|\nabla
w|^{2}=\int_{\Omega}(u-b)w+(b-\bar{u})\int_{\Omega}w\leq\|u-b\|_{L^{2}}\|w\|_{L^{2}}\leq
C_{p}\|u-b\|_{L^{2}}\|\nabla w\|_{L^{2}}.$

which shows

$\|\nabla w\|_{L^{2}}\leq C_{p}\|u-b\|_{L^{2}}.$ (5.39)

Next, we deduce from the third equation in (5.1) and the fact that
$\int_{\Omega}\Delta w=0$ that

$\begin{split}\int_{\Omega}|\Delta w|^{2}&=\int_{\Omega}(b-u)\Delta
w+(\bar{u}-b)\int_{\Omega}\Delta w\\\ &=-\int_{\Omega}(u-b)\Delta
w\leq\left(\int_{\Omega}(u-b)^{2}\right)^{\frac{1}{2}}\left(\int_{\Omega}|\Delta
w|^{2}\right)^{\frac{1}{2}},\end{split}$

which along with the $W^{2,2}$-elliptic estimate applied to (3.27) yields

$\|w\|_{W^{2,2}}\leq c_{80}\|\Delta w\|_{L^{2}}\leq c_{80}\|u-b\|_{L^{2}}.$

This together with (5.39) gives rise to (5.38). ∎

######  Lemma 5.6.

The classical solution of (5.37) satisfies

$\frac{d}{dt}\int_{\Omega}(u-b-b\ln\frac{u}{b})+\mu\int_{\Omega}(u-b)(u^{\theta}-b^{\theta})\leq\frac{b\chi^{2}}{2}\int_{\Omega}|\nabla
v|^{2}+\frac{b\xi_{1}^{2}C_{p}^{2}}{2}\int_{\Omega}(u-b)^{2}.$ (5.40)

###### Proof.

We use integration by parts to deduce from (5.37) that

$\begin{split}\frac{d}{dt}\int_{\Omega}(u-b-b\ln\frac{u}{b})&+b\int_{\Omega}\frac{|\nabla
u|^{2}}{u^{2}}+\mu\int_{\Omega}(u-b)(u^{\theta}-b^{\theta})\\\
&=b\chi\int_{\Omega}\frac{\nabla u\cdot\nabla
v}{u}-b\xi_{1}\int_{\Omega}\frac{\nabla u\cdot\nabla w}{u}.\end{split}$ (5.41)

Using Young’s inequality, one has

$b\chi\int_{\Omega}\frac{\nabla u\cdot\nabla
v}{u}\leq\frac{b}{2}\int_{\Omega}\frac{|\nabla
u|^{2}}{u^{2}}+\frac{b\chi^{2}}{2}\int_{\Omega}|\nabla v|^{2},$ (5.42)

and, due to (5.38) of Lemma 5.5,

$-b\xi_{1}\int_{\Omega}\frac{\nabla u\cdot\nabla
w}{u}\leq\frac{b}{2}\int_{\Omega}\frac{|\nabla
u|^{2}}{u^{2}}+\frac{b\xi_{1}^{2}}{2}\int_{\Omega}|\nabla
w|^{2}\leq\frac{b}{2}\int_{\Omega}\frac{|\nabla
u|^{2}}{u^{2}}+\frac{b\xi_{1}^{2}C_{p}^{2}}{2}\int_{\Omega}(u-b)^{2}.$ (5.43)

Then we substitute (5.42) and (5.43) into (5.41) to obtain (5.40). ∎

######  Lemma 5.7.

The global bounded classical solution of (5.37) fulfils

$\frac{d}{dt}\int_{\Omega}(v-b)^{2}+\int_{\Omega}(v-b)^{2}+d\int_{\Omega}|\nabla
v|^{2}\leq\left(1+\frac{\xi_{2}^{2}M_{0}^{2}C_{p}^{2}}{d}\right)\int_{\Omega}(u-b)^{2}.$
(5.44)

###### Proof.

From (5.37), we use integration by parts, the fact
$\|v(\cdot,t)\|_{L^{\infty}}\leq M_{0}$ in (3.1), the Hölder inequality and
Young’s inequality to estimate

$\begin{split}&\frac{d}{dt}\int_{\Omega}(v-b)^{2}+2\int_{\Omega}(v-b)^{2}+2d\int_{\Omega}|\nabla
v|^{2}\\\ &=-2\xi_{2}\int_{\Omega}v\nabla v\cdot\nabla
w+2\int_{\Omega}(u-b)(v-b)\\\ &\leq 2\xi_{2}M_{0}\|\nabla v\|_{L^{2}}\|\nabla
w\|_{L^{2}}+\int_{\Omega}(u-b)^{2}+\int_{\Omega}(v-b)^{2}\\\ &\leq
d\int_{\Omega}|\nabla
v|^{2}+\frac{\xi_{2}^{2}M_{0}^{2}}{d}\int_{\Omega}|\nabla
w|^{2}+\int_{\Omega}(u-b)^{2}+\int_{\Omega}(v-b)^{2},\end{split}$

which in conjunction with (5.38) shows (5.44). ∎

We are now ready to show the exponential decay of global bounded solutions.

######  Lemma 5.8.

Let $(u,v,w)$ be the global solution of (5.37) obtained in Theorem 1.1. If
$\mu>0$ satisfies (1.15), then $(u,v,w)$ decays exponentially to $(b,b,0)$:
for some $\sigma>0$,

$\|u(\cdot,t)-b\|_{L^{\infty}}+\|v(\cdot,t)-b\|_{L^{\infty}}+\|w(\cdot,t)\|_{W^{2,\infty}}\leq
C_{14}e^{-\frac{\sigma}{2(n+1)}t},\ \ \ t>0.$ (5.45)

###### Proof.

By the estimate for $M_{0}$ in (3.1) or (1.7), we first infer that

$M_{0}^{2}\leq
O(1)\left(1+\left(\frac{1}{\mu}\right)^{\frac{4+n}{\theta}}\right),$

and then, we find that our assumption on $\mu$ in (1.15) along with (1.14)
implies

$\mu>\frac{\left[\left(1+\frac{C^{2}_{p}\xi_{2}^{2}M_{0}^{2}}{d}\right)\frac{\chi^{2}}{d}+C_{p}^{2}\xi_{1}^{2}\right]}{2b^{\theta-2}}\Longleftrightarrow\mu>\left(\frac{d\chi^{2}+d^{2}C^{2}_{p}\xi_{1}^{2}+C^{2}_{p}\xi_{2}^{2}M_{0}^{2}}{2d^{2}a^{\frac{\theta-2}{\theta}}}\right)^{\frac{\theta}{2}}.$
(5.46)

Now, multiplying (5.44) by $\frac{b\chi^{2}}{2d}$ and adding the resulting
inequality to (5.40), we obtain

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}(u-b-b\ln\frac{u}{b})+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2}\right)+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2}\\\
&\leq-\mu\int_{\Omega}(u-b)(u^{\theta}-b^{\theta})+\frac{b}{2}\left[\left(1+\frac{C^{2}_{p}\xi_{2}^{2}M_{0}^{2}}{d}\right)\frac{\chi^{2}}{d}+C^{2}_{p}\xi_{1}^{2}\right]\int_{\Omega}(u-b)^{2}.\end{split}$
(5.47)

Notice, since $\theta\geq 1$, by examining monotonicity, one easily sees that

$\sup_{z\in(0,b)\cup(b,\infty)}\frac{(z-b)^{2}}{(z-b)(z^{\theta}-b^{\theta})}=\frac{1}{b^{\theta-1}}\Longleftrightarrow-(z-b)(z^{\theta}-b^{\theta})\leq-b^{\theta-1}(z-b)^{2},\
\ \forall z>0.$

Applying this inequality to (5.47), we immediately arrive at

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}(u-b-b\ln\frac{u}{b})+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2}\right)+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2}\\\
&\ \
+b^{\theta-1}\left[\mu-\frac{\left[\left(1+\frac{C_{p}^{2}\xi_{2}^{2}M_{0}^{2}}{d}\right)\frac{\chi^{2}}{d}+C_{p}^{2}\xi_{1}^{2}\right]}{2b^{\theta-2}}\right]\int_{\Omega}(u-b)^{2}\leq
0.\end{split}$ (5.48)

We first integrate (5.48) from $1$ to $t$ and then we use our derived
condition (5.46) and the elementary algebraic fact that
$z-b-b\ln\frac{z}{b}\geq 0$ for $z>0$ to obtain that

$\int_{1}^{t}\int_{\Omega}(u-b)^{2}<c_{81},\ \ \forall t\geq 1.$ (5.49)

Given (5.49), using the same arguments as in Lemma 5.3 for $u$, one can easily
show that

$\|u(\cdot,t)-b\|_{L^{\infty}}\to 0\ \ \mathrm{as}\ t\to\infty.$ (5.50)

Next, observing the following elementary fact that

$\lim_{z\rightarrow b}\frac{(z-b)^{2}}{z-b-b\ln\frac{z}{b}}=2b,$

we infer from (5.50) there exists $t_{2}>0$ such that

$b\left[u(\cdot,t)-b-b\ln\frac{u(\cdot,t)}{b}\right]\leq\left[u(\cdot,t)-b\right]^{2}\leq
3b\left[u(\cdot,t)-b-b\ln\frac{u(\cdot,t)}{b}\right],\ \forall t\geq t_{2}.$
(5.51)

Substituting (5.51) into (5.48), we end up with an important ODI:

$\begin{split}&\frac{d}{dt}\left(\int_{\Omega}(u-b-b\ln\frac{u}{b})+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2}\right)\\\
&\ \
+\sigma\left(\int_{\Omega}(u-b-b\ln\frac{u}{b})+\frac{b\chi^{2}}{2d}\int_{\Omega}(v-b)^{2}\right)\leq
0,\ \ \ \forall t\geq t_{2},\end{split}$ (5.52)

where, due to (5.46),

$\sigma:=\min\left\\{1,\ \ \
b^{\theta}\left[\mu-\frac{\left[\left(1+\frac{C^{2}_{p}\xi_{2}^{2}M_{0}^{2}}{d}\right)\frac{\chi^{2}}{d}+C^{2}_{p}\xi_{1}^{2}\right]}{2b^{\theta-2}}\right]\right\\}>0.$

Solving the ODI (5.52), we simply get the following exponential decay
estimate:

$\int_{\Omega}(u-b-b\ln\frac{u}{b})+\int_{\Omega}(v-b)^{2}\leq
c_{82}e^{-\sigma(t-t_{2})},\ \ \ \forall t\geq t_{2},$

which combined with (5.51) gives

$\|u(\cdot,t)-b\|_{L^{2}}+\|v(\cdot,t)-b\|_{L^{2}}\leq
c_{83}e^{-\frac{\sigma}{2}(t-t_{2})},\ \ \ \forall t\geq t_{2}.$ (5.53)

Now, it is easy to improve this $L^{2}$-convergence to
$L^{\infty}$-convergence as in Lemma 5.3. Indeed, by the boundedness of
$\|\nabla u\|_{L^{2n}}$ in (4.3) and the GN inequality, (5.53) entails

$\begin{split}\|u(\cdot,t)-b\|_{L^{\infty}}&\leq c_{84}\left(\|\nabla
u(\cdot,t)\|^{\frac{n}{n+1}}_{L^{2n}}\|u(\cdot,t)-b\|^{\frac{1}{n+1}}_{L^{2}}+\|u(\cdot,t)-b\|_{L^{2}}\right)\\\
&\leq c_{85}e^{-\frac{\sigma}{2(n+1)}t},\ \ \ \forall t\geq t_{2}.\end{split}$
(5.54)

Similarly, using the $W^{1,\infty}$-boundedness of $v$ guaranteed by Theorem
1.1, we use the GN inequality to improve the $L^{2}$-convergence of $v$ in
(5.53) to its $L^{\infty}$-convergence as follows:

$\begin{split}\|v(\cdot,t)-b\|_{L^{\infty}}\leq
c_{86}\|v(\cdot,t)\|_{W^{1,\infty}}^{\frac{n}{n+2}}\|v(\cdot,t)-b\|_{L^{2}}^{\frac{2}{n+2}}&\leq
c_{87}e^{-\frac{\sigma}{n+2}(t-t_{2})},\ \ \ \forall t\geq t_{2}.\end{split}$
(5.55)

As for the $W^{2,\infty}$-convergence of $w$, applying the
$W^{2,\infty}$-estimate to (3.27) and using (5.54), we see that $w$ decays to
zero exponentially in $W^{2,\infty}$-norm:

$\begin{split}\|w(\cdot,t)\|_{W^{2,\infty}}&\leq c_{88}\|\Delta
w\|_{L^{\infty}}\leq
c_{88}\left(\|u(\cdot,t)-b\|_{L^{\infty}}+|b-\bar{u}|\right)\\\ &\leq
2c_{88}\|u(\cdot,t)-b\|_{L^{\infty}}\leq c_{89}e^{-\frac{\sigma}{2(n+1)}t},\ \
\ \forall t\geq t_{2}.\end{split}$ (5.56)

Our desired exponential convergence (5.45) follows from (5.54), (5.55) and
(5.56). ∎

###### Proof of Theorem 1.3.

The respective of convergence and exponential convergence in (C1) and (C2) of
Theorem 1.3 have been fully contained in Lemmas 5.3, 5.4, and 5.8. ∎

Acknowledgment. H. Y. Jin was supported by the Guangdong Basic and Applied
Basic Research Foundation (No. 2020A1515010140and 2020B1515310015),Guangzhou
Science and Technology Pro- gram (No. 202002030363), NSF of China (No.
11871226 and No.12026608), and the Fundamental Research Funds for the Central
Universities. T. Xiang was funded by the NSF of China (No. 12071476 and
11871226) and the Research Funds of Renmin University of China (No.
2018030199).

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