# Asymptotic profile of a two-dimensional chemotaxis–Navier–Stokes system with
singular sensitivity and logistic source

Peter Y. H. Pang111Email<EMAIL_ADDRESS>
Department of Mathematics, National University of Singapore
10 Lower Kent Ridge Road, Republic of Singapore 119076
Yifu Wang222Corresponding author. Email<EMAIL_ADDRESS>
School of Mathematics and Statistics, Beijing Institute of Technology
Beijing 100081, People’s Republic of China
Jingxue Yin333Email<EMAIL_ADDRESS>
School of Mathematics, South China Normal University
Guangzhou, Guangdong 510631, People’s Republic of China

###### Abstract

The chemotaxis–Navier–Stokes system

$\left\\{\begin{array}[]{ll}n_{t}+u\cdot\nabla n=\triangle
n-\chi\nabla\cdotp\left(\displaystyle\frac{n}{c}\nabla c\right)+n(r-\mu n),\\\
c_{t}+u\cdot\nabla c=\triangle c-nc,\\\ u_{t}+(u\cdot\nabla)u=\Delta u+\nabla
P+n\nabla\phi,\\\ \nabla\cdot u=0,\end{array}\right.$

is considered in a bounded smooth domain $\Omega\subset\mathbb{R}^{2}$, where
$\phi\in W^{1,\infty}(\Omega)$, $\chi>0$, $r\in\mathbb{R}$ and $\mu>0$ are
given parameters. It is shown that there exists a value
$\mu_{*}(\Omega,\chi,r)\geq 0$ such that whenever
$\mu>\mu_{*}(\Omega,\chi,r)$, the global-in-time classical solution to the
system is uniformly bounded with respect to $x\in\Omega$. Moreover, for the
case $r>0$, $(n,c,\frac{|\nabla c|}{c},u)$ converges to
$(\frac{r}{\mu},0,0,0)$ in $L^{\infty}(\Omega)\times L^{\infty}(\Omega)\times
L^{p}(\Omega)\times L^{\infty}(\Omega)$ for any $p>1$ exponentially as
$t\rightarrow\infty$, while in the case $r=0$, $(n,c,\frac{|\nabla c|}{c},u)$
converges to $(0,0,0,0)$ in $(L^{\infty}(\Omega))^{4}$ algebraically. To the
best of our knowledge, these results provide the first precise information on
the asymptotic profile of solutions

Keywords: Chemotaxis, Navier–Stokes, singular sensitivity, asymptotic profile.

AMS Classification (2010): 35K55, 35Q30, 35B40, 35Q92, 76D05, 92C17.

##  1 Introduction

In their seminal work ([14, 15]), using cross-diffusive parabolic PDE systems,
Keller and Segel studied the phenomenon of the emergence of spatial structures
in biological systems through chemically induced processes. In particular,
they looked at situations where componenets of the biological system were able
to actively secrete a chemoattractive signal which then directed the movements
of components in the system; or where, instead of having the ability to
produce such signals, components of the system simply followed and consumed a
chemical nutrient produced externally. A prototypical example of the former is
the dictyostelium discoideum colony, while that of the latter is an E. coli
population. The movement thus induced by either a chemical signal or nutrient
is called chemotaxis, and the corresponding model of the latter type is
sometimes called a chemotaxis–consumption model. Often, such chemotactic
movements take place in a fluid environment, and experimental findings and
analytical studies have revealed the remarkable effects of chemotaxis–fluid
interaction on the overall behavior of the respective chemotaxis systems, such
as the prevention of blow-up and improvement of efficiency of mixing ([8, 16,
17, 29, 37]).

In this paper, we are concerned with the chemotaxis–consumption system coupled
with the incompressible Navier–Stokes equations

$\left\\{\begin{array}[]{ll}n_{t}+u\cdot\nabla n=\triangle
n-\chi\nabla\cdotp\left(\displaystyle\frac{n}{c}\nabla
c\right)+f(n),&x\in\Omega,t>0,\\\ c_{t}+u\cdot\nabla c=\triangle
c-nc,&x\in\Omega,t>0,\\\ u_{t}+(u\cdot\nabla)u=\Delta u+\nabla
P+n\nabla\phi,&x\in\Omega,t>0,\\\ \nabla\cdot u=0,&x\in\Omega,t>0,\\\
\end{array}\right.$ (1.1)

describing the biological population density $n$, the chemical signal
concentration $c$, the incompressible fluid velocity $u$ and the associated
pressure $P$ of the fluid flow in the physical domain
$\Omega\subset\mathbb{R}^{N}$. It is assumed that $n$ and $c$ diffuse randomly
as well as are transported by the fluid, with a bouyancy effect on $n$ through
the presence of a given gravitational potential $\phi$. Further, it is assumed
that the chemotactic stimulus is perceived in accordance with the
Weber–Fechner Law ([34, 40, 52]) which states that subjective sensation is
proportional to the logarithm of the stimulus intensity, in other words, the
population $n$ partially direct their movement toward increasing
concentrations of the chemical nutrient $c$ that they consume with the
logarithmic sensitivity. In addition, on the considered time scales of cell
migration, we allow for population growth to take place, through the term
$f(n)=rn-\mu n^{2}$ with the effective growth rate $r\in\mathbb{R}$ and
strength of the overcrowding effect $\mu>0$; we note that $r=0$ is allowed and
has indeed been argued for in certain models ([12, 16]).

The system (1.1) appears to generate interesting, non-trivial dynamics.
However, to the best of our knowledge, no analytical result is available yet
which rigorously describes the qualitative behavior of such solutions. This
may be due to the circumstance that (1.1) joins two subsystems which are far
from being fully understood even when decoupled from each other. Indeed, (1.1)
contains the Navier–Stokes equations which themselves do not admit a complete
existence and regularity theory ([42]).

At the same time, by setting $u\equiv 0$ in (1.1), we arrive at the following
chemotaxis–consumption model

$\left\\{\begin{array}[]{ll}n_{t}=\triangle
n-\chi\nabla\cdotp(\frac{n}{c}\nabla c),\\\ c_{t}=\triangle
c-nc,\end{array}\right.$ (1.2)

where population growth has been ignored, which was introduced by Keller and
Segel ([15]) to describe the collective behaviour of the bacteria E. coli set
in one end of a capillary tube featuring a gradient of nutrient concentration
observed in the celebrated experiment of Adler ([1]). Later, this model was
also employed to describe the dynamical interactions between vascular
endothelial cells and vascular endothelial growth factor (VEGF) during the
initiation of tumor angiogenesis (see [9, 26]). It has already been
demonstrated that the logarithmic sensitivity featured in (1.2) renders a
significant degree of complexity in the system; in particular, it plays an
indispensable role in generating wave-like solutions without any type of cell
kinetics ([12, 14, 32, 33, 40]), which is a prominent feature in the Fisher
equation ([18]).

In comparison with (1.2), the related chemotaxis system

$\left\\{\begin{array}[]{ll}n_{t}=\triangle
n-\chi\nabla\cdotp(\frac{n}{c}\nabla c)+f(n),\\\ c_{t}=\triangle
c-c+n,\end{array}\right.$ (1.3)

where the chemical signal $c$ is actively secreted by the bacteria rather than
consumed (see [2, 12]), has been more extensively studied. It is observed that
the chemical signal production mechanism in the $c-$equation inhibits the
tendency of $c$ to take on small values, and thereby the singularity in the
sensitivity function is mitigated. Accordingly, for such higher dimensional
systems with reasonably smooth but arbitrarily large data, the global
existence of bounded smooth solutions can be achieved. Indeed, global
existence and boundedness of classical solutions to (1.3) without source terms
is guaranteed if $\chi\in(0,\sqrt{\frac{2}{N}})$ ([11, 45]), or if
$N=2,\chi\in(0,\chi_{0})$ with some $\chi_{0}>1.015$ ([22]), while certain
generalized solutions have been constructed for general $\chi>0$ in the two-
dimensional radially symmetric case ([35, 45]). Moreover, without any symmetry
hypothesis, Winkler and Lankeit established the global solvability of
generalized solutions for the cases $\chi<\infty,N=2$; $\chi<\sqrt{8},N=3$;
and $\chi<\frac{N}{N-2},N\geq 4$ ([24]).

Furthermore, in accordance with known results for the classical Keller–Segel
chemotaxis model (see [21, 44, 46] for example), the presence of the logistic
source term $f(n)=n(r-\mu n)$ in (1.3) can inhibit the tendency toward
explosions of cells at least under some restrictions on certain parameters.
Indeed, it is known that (1.3) with $N=2$ possesses a global classical
solution $(n,c)$ for any $r\in\mathbb{R},\chi,\mu>0$, and $(n,c)$ is globally
bounded if $r>\frac{\chi^{2}}{4}$ for $0<\chi\leq 2$ or $r>\chi-1$ for
$\chi>2$ ([53]). Moreover, $(n,c)$ exponentially converges to
$(\frac{r}{\mu},\frac{r}{\mu})$ in $L^{\infty}(\Omega)$ provided that $\mu>0$
is sufficiently large ([56]). As for the higher dimensional cases ($N\geq 2$),
the global very weak solution of (1.3) with $f(n)=rn-\mu n^{k}$ is constructed
when $k,\chi$ and $r$ fulfill a certain condition. In addition, when $N=2$ or
$3$, this solution is global bounded provided $\frac{r}{\mu}$ and the initial
data $\|n_{0}\|_{L^{2}},\|\nabla c_{0}\|_{L^{4}}$ are suitably small ([55]).

In contrast to (1.3), system (1.2) is more challenging due to the combination
of the consumption of $c$ with the singular chemotaxis sensitivity of $n$.
Intuitively, the absorption mechanism in the $c-$equation of (1.2), which
induces the preference for small values of $c$, considerably intensifies the
destabilizing potential of singular sensitivity in the $n-$equation. Up to
now, it seems that only limited results on global classical solvability in the
spatial two-dimensional case are available. In fact, only recently have
certain global generalized solutions to (1.2) been constructed for general
initial data in [20, 47, 50], whereas with respect to global classical
solvability, it has only been shown for some small initial data (see [41,
48]). In particular, Winkler ([48]) showed that the global classical solutions
to (1.2) in bounded convex two-dimensional domains exist and converge to the
homogeneous steady state under an essentially explicit smallness condition on
$n_{0}$ in $L\log L(\Omega)$ and $\nabla\ln c_{0}$ in $L^{2}(\Omega)$.

We would, however, like to note that numerous variants of (1.2), such as those
involving nonlinear diffusion, logistic-type cell kinetics and saturating
signal production ([10, 13, 19, 23, 25, 28, 38, 54]), have been studied. For
example, the authors of [54] proved that the particular version of (1.2) by
adding $f(n)=rn-\mu n^{k}~{}(r>0,\mu>0,k>1)$ into the $n$-equation admits a
global classical solution $(n,c)$ in the bounded domain
$\Omega\subset\mathbb{R}^{N}$ if $k>1+\frac{N}{2}$, and in the two-dimensional
setting, $(n,c,\frac{|\nabla
c|}{c})\rightarrow((\frac{r}{\mu})^{\frac{1}{k-1}},0,0)$ for sufficiently
large $\mu$. In particular, it is shown in the recent paper [19] that (1.2)
with logistic source $f(n)=rn-\mu n^{2}~{}(r\in\mathbb{R},\mu>0)$ possesses a
unique global classical solution if $0<\chi<\sqrt{\frac{2}{N}}$,
$\mu>\frac{N-2}{2N}$, and a globally bounded solution only in one dimension
for any $\chi>0,\mu>0$. Also, the author of [39] showed that if $\mu>\mu_{0}$
with some $\mu_{0}=\mu_{0}(\Omega,\chi)>0$ then the corresponding classical
solution is globally bounded, and $(n,c,\frac{|\nabla
c|}{c})\rightarrow(\frac{r_{+}}{\mu},\lambda,0)$ with
$\lambda\in[0,\frac{1}{|\Omega|}\int_{\Omega}c_{0})$ in
$(L^{\infty}(\Omega))^{3}$ as $t\rightarrow\infty$. Of course, this leaves
open the possibility of blow-up of solutions when $\mu$ is positive but small.
Anyhow, it have been shown in [49] that when $\mu>0$ is suitably small, the
strongly destablizating action of chemotactic cross–diffusion may lead to the
occurrence of solutions which attain possibly finite but arbitrarily large
values.

Coming back to our chemotaxis–consumption–fluid model (1.1), as we have
already pointed out, very little seems to be known regarding the qualitative
behavior of solutions ([4, 5, 6]). In fact, we are aware of one result only
which is concerned with the asymptotic behavior and eventual regularity of
solutions to the Stokes–variant of (1.1). Namely, it is shown in [4] that for
small initial mass $\int_{\Omega}n_{0}$, the corresponding system upon
neglection of $u\cdot\nabla u$ and $f(n)$ in (1.1) possesses at least one
global generalized solutions, which will become smooth after some waiting time
and stabilize toward the steady state
$(\frac{1}{|\Omega|}\int_{\Omega}n_{0},0,0)$ with respect to the topology of
$(L^{\infty}(\Omega))^{3}$. Since the presence of the fluid interaction does
not have any regularizing effect on the large time behavior, it is expected
that instead of the small restriction on the initial data, the quadratic
degradation may have a substantial regularizing effect on the dynamic behavior
of solutions to (1.1).

The goal of the present work is to give the asymptotic profile in time of
solutions to (1.1) in the two-dimensional case. In order to state our main
results, we shall impose on (1.1) the boundary conditions

$\nabla n\cdot\nu=\nabla
c\cdot\nu=0\quad\hbox{and}~{}~{}u=0~{}~{}\hbox{for}~{}~{}x\in\partial\Omega,$
(1.4)

and initial conditions

$n(x,0)=n_{0}(x),~{}c(x,0)=c_{0}(x),~{}u(x,0)=u_{0}(x)~{}~{}\hbox{for}~{}~{}x\in\Omega.$
(1.5)

Throughout this paper, it is assumed that

$\left\\{\begin{array}[]{l}n_{0}\in C^{0}(\bar{\Omega}),~{}n_{0}\geq
0~{}\hbox{and}~{}n_{0}\not\equiv 0~{}\hbox{in}~{}\Omega,\\\ c_{0}\in
W^{1,\infty}(\Omega),c_{0}>0~{}\hbox{in}~{}\bar{\Omega}~{}~{}\hbox{as well
as}\\\ u_{0}\in D(A^{\beta})\;\hbox{for
all}\;\beta\in(\frac{1}{2},1)\end{array}\right.$ (1.6)

with $A$ denoting the Stokes operator $A=-\mathcal{P}\Delta$ with domain
$D(A):=W^{2,2}(\Omega;\mathbb{R}^{2})\cap
W^{1,2}_{0}(\Omega;\mathbb{R}^{2})\cap L_{\sigma}^{2}(\Omega)$, where
$L_{\sigma}^{2}(\Omega):=\\{\varphi\in
L^{2}(\Omega;\mathbb{R}^{2})|\nabla\cdot\varphi=0\\}$ and $\mathcal{P}$ stands
for the Helmholtz projection of $L^{2}(\Omega)$ onto $L_{\sigma}^{2}(\Omega)$.

Within this framework, by straightforward adaptation of arguments in [19] with
only some necessary modifications, one can see that the problem (1.1), (1.4),
(1.5) admits a global classical solution $(n,c,u,P)$ whenever $\chi\in(0,1)$,
$r\in\mathbb{R}$ and $\mu>0$, which is unique up to addition of constants in
the pressure variable $P$, and satisfies $n>0$, $c>0$ in
$\Omega\times[0,\infty)$. The first of our main results is concerned with the
global boundedness of the solution as well as its asymptotic behavior.

######  Theorem 1.1.

Let $f(n)=rn-\mu n^{2}$, $r\in\mathbb{R}$, $\mu>0$ and $\phi\in
W^{1,\infty}(\Omega)$, and suppose that $(n_{0},c_{0},u_{0})$ satisfy (1.6).
If $(n,c,u,P)$ denotes the corresponding global classical solution to (1.1),
(1.4), (1.5), then there exists a value $\mu_{0}:=\mu_{0}(\Omega,\chi,r)\geq
0$ with $\mu_{0}(\Omega,\chi,0)=0$ such that whenever $\mu>\mu_{0}$, $(n,c,u)$
is global bounded,

$\|n(\cdot,t)-\frac{r_{+}}{\mu}\|_{L^{\infty}(\Omega)}\rightarrow
0,~{}~{}\|\frac{\nabla c}{c}(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow
0,~{}~{}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow 0~{}~{}$

and when $r>0$,

$~{}~{}\|c(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow 0~{}~{}~{}~{}$

as $t\rightarrow\infty$.

As indicated in the above discussion, we need to introduce new ideas to show
how the regularizing effect of the quadratic degradation in the
chemotaxis–fluid model (1.1) can counterbalance the strongly destablizating
action of chemotactic cross–diffusion caused by the combination of the
consumption of $c$ with the singular chemotaxis sensitivity of $n$.
Specifically, we develop the conditional energy functional method in [48] to
show the global boundedness of solutions in the case of $r>0$, in which the
key point is to verify that

$\mathcal{F}(n,w):=\int_{\Omega}H(n)+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla
w|^{2},~{}~{}w:=-\ln(\frac{c}{\|c_{0}\|_{L^{\infty}(\Omega)}})$ (1.7)

with $H(s):=s\ln\frac{\mu s}{er}+\frac{r}{\mu}$ constitutes an energy
functional in the sense that $\mathcal{F}(n,w)$ is non-increasing in time
whenever $\mu$ is appropriately large relative to $r$ (see Lemma 3.2). Indeed,
from (3.13), one can obtain the global bound of $\int_{\Omega}n|\ln n|dx$ and
$\int_{\Omega}|\nabla w|^{2}dx$, which then serves as a starting point to
derive the uniform bound of $\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ via the
Neumann heat semigroup estimates. Furthermore, by making appropriate use of
the dissipative information expressed in (3.13), we can establish the
convergence result asserted in Theorem 1.1. It is noted that compared to that
of the case $r>0,\mu>0$, the proof of Theorem 1.1 in the case of $r\leq
0,\mu>0$ involves a more delicate analysis. In fact, unlike in the case
$r=\mu=0$ or $r>0,\mu>0$, (1.1) with $r\leq 0,\mu>0$ seems to lack the
favorable structure that facilitates such conditional energy-type
inequalities. Taking full advantage of the decay information on $n$ in
$L^{1}-$norm expressed in (2.2), our approach toward Theorem 1.1 is to
construct the quantity

$\mathcal{F}(n,w):=\int_{\Omega}n(\ln
n+a)+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla w|^{2}$ (1.8)

with parameter $a>0$ determined below (see (3.49)). Unlike in the case of
$r>0$, $\mathcal{F}(n,w)$ does not enjoy monotonicity property, it however
satisfies a favorable non-homogeneous differential inequality (3.55) in the
sense that it can provide us a priori information on solution such as the
global bound of $\int_{\Omega}n|\ln n|dx$ and $\int_{\Omega}|\nabla w|^{2}dx$
(see Lemma 3.4), as well as
$\displaystyle\lim_{t\rightarrow\infty}\int_{\Omega}|\nabla w(\cdot,t)|^{2}=0$
(see (3.61)).

As an important step to understand the model (1.1) more comprehensively, we
shall consider the convergence rate of its classical solutions in the form of
the following result:

######  Theorem 1.2.

Let the assumptions of Theorem 1.1 hold and $r>0$. Then one can find
$\mu_{*}(\chi,\Omega,r)>0$ such that if $\mu>\mu_{*}(\chi,\Omega,r)$, the
classical solution of (1.1), (1.4), (1.5) presented in Theorem 1.1 satisfies

$\|n(\cdot,t)-\frac{r}{\mu}\|_{L^{\infty}(\Omega)}\rightarrow
0,~{}~{}\|c(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow
0,~{}~{}~{}~{}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow 0$

as well as

$\|\frac{\nabla c}{c}(\cdot,t)\|_{L^{p}(\Omega)}\rightarrow 0$

for all $p>1$ exponentially as $t\rightarrow\infty$.

This implies that suitably large $\mu$ relative to $r$ enforces asymptotic
stability of the corresponding constant equilibria of (1.1); however, the
optimal lower bound on $\frac{\mu}{r}$ seems yet lacking. The main ingredient
of the our approach toward Theorem 1.2 involves a so-called self-map-type
reasoning. More precisely, making use of the convergence properties of
$(n,\frac{|\nabla c|}{c})$ asserted in Theorem 1.1, we prove by a self-map-
type reasoning that whenever $\mu$ is suitably large compared with $r$,

$(n(\cdot,t)-\frac{r}{\mu},c(\cdot,t),u)\longrightarrow(0,0,0)~{}~{}\hbox{and}~{}~{}\frac{|\nabla
c|}{c}(\cdot,t)\longrightarrow 0$

in $(L^{\infty}(\Omega))^{3}$ and $L^{6}(\Omega)$ exponentially as
$t\rightarrow\infty$, respectively (see Lemma 4.2).

As aforementioned, the limit case $r=0$ becomes relevant in several
applications. In this limiting situation, the total cell population can
readily be seen to decay in the large time limit (cf. Lemma 2.3 below). As a
consequence, we can obtain the decay properties of solutions, namely that the
decay on $n$ in $L^{1}$ actually occurs in $L^{\infty}$, and also for $c$.
More precisely, our result reads as follows:

######  Theorem 1.3.

Let the assumptions of Theorem 1.1 hold and $r=0$. Then the classical solution
of (1.1), (1.4), (1.5) from Theorem 1.1 satisfies

$(n,c,\frac{|\nabla c|}{c},u)\longrightarrow(0,0,0,0)$

in $(L^{\infty}(\Omega))^{4}$ algebraically as $t\rightarrow\infty$.

The result indicates that structure generating dynamics in the spatially two-
dimensional version of (1.1), (1.4), (1.5), if at all, occur on intermediate
time scales rather than in the sense of a stable large-time pattern formation
process. Apparently, it leaves open the questions whether the more colorful
large time behavior can appear in the three-dimensional version of (1.1).

Our approach toward Theorem 1.3 uses an alternative method, which, at its
core, is based on the argument that the $L^{\infty}$-norm of $n$ can be
controlled from above by appropriate multiples of $\frac{1}{t+1}$. This
results from a suitable variation-of-constants representation of $n$, by which
and in view of the decay information on $|\nabla w|$ in $L^{\infty}(\Omega)$,
the $L^{1}$ decay information on $u$ from (2.2) can be turned into the
$L^{\infty}$-norm of $n$ (see Lemma 4.3). As a consequence, by comparison
argument, we have a pointwise upper estimate for $w$ as well as a lower
estimate for $v$ (see Lemma 4.4). Using $L^{p}-L^{q}$ estimates for the
Neumann heat semigroup $(e^{t\Delta})_{t>0}$, we then successively show that
$\|\nabla w\|_{L^{\infty}}$ and $\|n\|_{L^{\infty}(\Omega)}$ can be controlled
by appropriate multiples of $\frac{1}{t+1}$ from above and below, respectively
(see Lemma 4.5). These a priori estimates allow us to get the pointwise lower
estimate for $w$ as well as the upper estimate for $c$, which complement the
lower bound for $c$ previously obtained, and thereby prove that $c$ actually
decays algebraically.

##  2 Preliminaries

In this section, we begin by recalling the important $L^{p}-L^{q}$ estimates
for the Neumann heat semigroup $(e^{t\Delta})_{t>0}$ on bounded domains (see
[7, 43]).

######  Lemma 2.1.

Let $(e^{t\Delta})_{t>0}$ denote the Neumann heat semigroup in the bounded
domain $\Omega\subset\mathbb{R}^{n}$ and $\lambda_{1}>0$ denote the first
nonzero eigenvalue of $-\Delta$ in $\Omega$ under the Neumann boundary
condition. Then there exists $c_{i}>0$ ($i=1,2,3$) such that for all $t>0$,

$(i)$ If $1\leq q\leq p\leq\infty$, then for all $\omega\in L^{q}(\Omega)$,

$\|\nabla e^{t\Delta}\omega\|_{L^{p}(\Omega)}\leq
c_{1}\left(1+t^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|\omega\|_{L^{q}(\Omega)};$

$(ii)$ If $2\leq q\leq p<\infty$, then for all $\omega\in W^{1,q}(\Omega)$,

$\|\nabla e^{t\Delta}\omega\|_{L^{p}(\Omega)}\leq
c_{2}\left(1+t^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|\nabla\omega\|_{L^{q}(\Omega)};$

$(iii)$ If $1\leq q\leq p\leq\infty$ or $1<q<\infty$ and $p=\infty$, then for
all $\omega\in(L^{q}(\Omega))^{n}$,

$\|e^{t\Delta}\nabla\cdot\omega\|_{L^{p}(\Omega)}\leq
c_{3}\left(1+t^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\right)e^{-\lambda_{1}t}\|\omega\|_{L^{q}(\Omega)}.$

######  Lemma 2.2.

(Gagliardo–Nirenberg inequality) Let $\Omega\subset\mathbb{R}^{2}$ be a
bounded Lipschitz domain. Then i) there is $K_{1}>0$ such that

$\|\nabla\varphi\|^{4}_{L^{4}(\Omega)}\leq
K_{1}\|\Delta\varphi\|^{2}_{L^{2}(\Omega)}\|\nabla\varphi\|^{2}_{L^{2}(\Omega)}$

for all $\varphi\in W^{2,2}(\Omega)$ fulfilling
$\frac{\partial\varphi}{\partial\nu}|_{\partial\Omega}=0$;

ii) there is $K_{2}>0$ such that

$\|\varphi\|^{3}_{L^{3}(\Omega)}\leq
K_{2}\|\varphi\|^{2}_{W^{1,2}(\Omega)}\|\varphi\|_{L^{1}(\Omega)}$

for all $\varphi\in W^{1,2}(\Omega)$.

In order to derive some essential estimates, it would be more convenient to
deal with a nonsingular chemotaxis term of the form $\nabla\cdotp(n\nabla w)$
instead of $\nabla\cdotp(\frac{n}{c}\nabla c)$ in (1.1). To this end, we
employ the following transformation as in [19, 23, 47]:

$w:=-\ln(\frac{c}{\|c_{0}\|_{L^{\infty}(\Omega)}}),$

whereupon $0\leq w\in C^{0}(\bar{\Omega}\times(0,\infty))\cap
C^{2,1}(\bar{\Omega}\times(0,\infty))$, and the problem (1.1), (1.4), (1.5)
transforms to

$\left\\{\begin{array}[]{ll}n_{t}+u\cdot\nabla n=\triangle
n+\chi\nabla\cdotp(n\nabla w)+n(r-\mu n),&x\in\Omega,t>0,\\\
w_{t}+u\cdot\nabla w=\triangle w-|\nabla w|^{2}+n,&x\in\Omega,t>0,\\\
u_{t}+(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,t>0,\\\
\nabla\cdot u=0,&x\in\Omega,t>0,\\\ \nabla n\cdot\nu=\nabla w\cdot\nu=0,\quad
u=0,&x\in\partial\Omega,t>0,\\\
n(x,0)=n_{0}(x),~{}w(x,0)=-\ln(\frac{c_{0}(x)}{\|c_{0}\|_{L^{\infty}(\Omega)}}),~{}~{}u=u_{0}(x),&x\in\Omega.\end{array}\right.$
(2.1)

Let us first recall some basic but important information about $(n,w)$ due to
the presence of the quadratic degradation term in the first equation of (2.1).

######  Lemma 2.3.

The classical solution $(n,w,u,P)$ of (2.1) satisfies

(i)
$\displaystyle\lim\sup_{t\rightarrow\infty}\|n(\cdot,t)\|_{L^{1}(\Omega)}\leq\frac{|\Omega|r_{+}}{\mu};$

(ii)$\displaystyle\int^{t}_{t_{0}}\|n(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds\leq\frac{r_{+}}{\mu}\int^{t}_{t_{0}}\|n(\cdot,s)\|_{L^{1}(\Omega)}ds+\frac{1}{\mu}\|n(\cdot,t_{0})\|_{L^{1}(\Omega)}$
for all $t>t_{0}$;

(iii) $\displaystyle\int^{t}_{t_{0}}\int_{\Omega}|\nabla
w|^{2}dxds\leq\int_{\Omega}w(x,t_{0})dx+\int^{t}_{t_{0}}\|n(\cdot,s)\|_{L^{1}(\Omega)}ds$
for all $t>t_{0}$.

In particular, if $r\leq 0$, then

$\|n(\cdot,t)\|_{L^{1}(\Omega)}\leq\displaystyle\frac{|\Omega|}{\mu(t+\gamma)}~{}~{}~{}\mbox{for
all}~{}~{}t>t_{0}$ (2.2)

with $\gamma=\displaystyle\frac{|\Omega|}{\mu\int_{\Omega}n_{0}(x)dx}$.

Proof. Integrating the first equation in (2.1) and using the Cauchy–Schwarz
inequality, we get

$\begin{array}[]{rl}\displaystyle\frac{d}{dt}\int_{\Omega}n=&\displaystyle
r\int_{\Omega}n-\mu\int_{\Omega}n^{2}\leq\displaystyle
r_{+}\int_{\Omega}n-\frac{\mu}{|\Omega|}(\int_{\Omega}n)^{2}\end{array}$ (2.3)

which yields (i) readily. By the time integration of (2.3) over $(t_{0},t)$,
we get (ii) immediately. In addition, from the second equation in (2.1),
$\nabla\cdot u=0$ and $u=0$ on $\partial\Omega$, it follows that

$\displaystyle\frac{d}{dt}\int_{\Omega}w=-\displaystyle\int_{\Omega}|\nabla
w|^{2}+\int_{\Omega}n,$ (2.4)

and thus establishes (iii).

When $r\leq 0$, it follows from (2.3) that

$\displaystyle\frac{d}{dt}\int_{\Omega}n\leq\displaystyle-\frac{\mu}{|\Omega|}(\int_{\Omega}n)^{2}$
(2.5)

which then yields (2.2) by the time integration.

In order to make use of the spatio-temporal properties provided by Lemma
2.3(ii) to estimate the ultimate bound of $\int_{\Omega}|\nabla u|^{2}$, we
shall utilize the following elementary lemma (see Lemma 3.4 of [51]):

######  Lemma 2.4.

Let $t_{0}\geq 0,T\in(t_{0},\infty]$, $a>0$ and $b>0$, and suppose that the
nonnegative function $h\in L^{1}_{loc}(\mathbb{R})$ satisfies
$\int^{t+1}_{t}h(s)ds\leq b$ for all $t\in[t_{0},T]$. If $y\in
C^{0}([t_{0},T))\cap C^{1}([t_{0},T))$ has the property that

$y^{\prime}(t)+ay(t)\leq h(t)~{}~{}\hbox{for all}~{}~{}t\in(t_{0},T),$

then

$y(t)\leq e^{-a(t-t_{0})}y(t_{0})+\frac{b}{1-e^{-a}}~{}~{}\hbox{for all
}~{}~{}t\in[t_{0},T).$

With Lemmas 2.3 and 2.4 at hand, we can employ the standard energy inequality
associated with the fluid evolution system in (2.1) to derive some boundedness
results for $u$.

######  Lemma 2.5.

For the global classical solution $(n,w,u)$ of (2.1), we have

i) if $r>0$, then

$\displaystyle\displaystyle\lim\sup_{t\rightarrow\infty}\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}\leq\frac{3(1+r)|\Omega|}{\mu}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{r}{\mu}$
(2.6)

as well as

$\displaystyle\displaystyle\lim\sup_{t\rightarrow\infty}\int^{t+1}_{t}\|\nabla
u(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds\leq\frac{5(1+r)|\Omega|}{\mu}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{r}{\mu}$
(2.7)

with Poincaré constant $C_{P}>0$.

ii) if $r\leq 0$, then

$\displaystyle\int_{\Omega}|u(\cdot,t)|^{2}\leq\|u(\cdot,t_{0})\|^{2}_{L^{2}(\Omega)}e^{-\frac{C_{p}}{2}(t-t_{0})}+\frac{2|\Omega|}{\mu^{2}}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{1}{t_{0}+\gamma}~{}~{}~{}~{}~{}\hbox{for
all}~{}~{}t>t_{0}$ (2.8)

as well as

$\displaystyle\int^{t+1}_{t}\|\nabla u(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds$
$\displaystyle\leq$
$\displaystyle\|u(\cdot,t_{0})\|^{2}_{L^{2}(\Omega)}e^{-\frac{C_{p}}{2}(t-t_{0})}+\frac{4|\Omega|}{\mu^{2}}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{1}{t_{0}+\gamma}~{}~{}~{}\hbox{for
all}~{}t>t_{0}.$ (2.9)

Proof. i) According to the Poincaré inequality, one can find some constant
$C_{p}>0$ such that

$C_{p}\int_{\Omega}|u|^{2}\leq\int_{\Omega}|\nabla u|^{2}.$

Testing the third equation in (2.1) by $u$ and using the Hölder inequality, we
obtain

$\displaystyle\frac{d}{dt}\int_{\Omega}|u|^{2}+C_{p}\int_{\Omega}|u|^{2}+\int_{\Omega}|\nabla
u|^{2}$ $\displaystyle\leq 2\int_{\Omega}n\nabla\phi\cdot u$
$\displaystyle\leq
2\|\nabla\phi\|_{L^{\infty}(\Omega)}\|n\|_{L^{2}(\Omega)}\|u\|_{L^{2}(\Omega)}$
$\displaystyle\leq\frac{C_{p}}{2}\|u\|_{L^{2}(\Omega)}^{2}+\frac{2}{C_{p}}\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\|n\|_{L^{2}(\Omega)}^{2},$

due to $u|_{\partial\Omega}=0$ and $\nabla\cdot u=0$.

Writing
$h(t)=\frac{2}{C_{p}}\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\|n(\cdot,t)\|_{L^{2}(\Omega)}^{2}$,
we see that $y(t):=\int_{\Omega}|u(\cdot,t)|^{2}$ satisfies

$\displaystyle y^{\prime}(t)+\frac{C_{p}}{2}y(t)+\int_{\Omega}|\nabla
u(\cdot,t)|^{2}\leq h(t)~{}~{}~{}~{}~{}\hbox{for all}~{}~{}t>0.$ (2.10)

In view of Lemma 2.3 (i) and (ii), we know that

$\displaystyle\displaystyle\lim\sup_{t\rightarrow\infty}\int^{t+1}_{t}h(s)ds\leq\frac{2}{C_{p}}\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\frac{(1+r)|\Omega|}{\mu}\frac{r}{\mu}.$
(2.11)

An application of Lemma 2.4 thus shows that there exists positive $t_{0}>0$
such that

$\int_{\Omega}|u(\cdot,t)|^{2}\leq\|u(\cdot,t_{0})\|^{2}_{L^{2}(\Omega)}e^{-\frac{C_{p}}{2}(t-t_{0})}+\frac{3(1+r)|\Omega|}{\mu}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{r}{\mu}~{}~{}~{}~{}~{}\hbox{for
all}~{}~{}t>t_{0}$

and thereby verifies (2.6). Thereafter, again thanks to (2.11), an integration
of (2.10) in time yields (2.7).

ii) In view of (2.2), we have

$\displaystyle\int^{t+1}_{t}h(s)ds\leq\frac{2}{C_{p}}\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\frac{|\Omega|}{\mu^{2}}\frac{1}{t+\gamma},$
(2.12)

whereupon Lemma 2.4 guarantees that

$\int_{\Omega}|u(\cdot,t)|^{2}\leq\|u(\cdot,t_{0})\|^{2}_{L^{2}(\Omega)}e^{-\frac{C_{p}}{2}(t-t_{0})}+\frac{2|\Omega|}{\mu^{2}}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{1}{t_{0}+\gamma}~{}~{}~{}~{}~{}\hbox{for
all}~{}~{}t>t_{0}.$

This precisely warrants (2.8), and thereby in turn yields (2.5) after
integrating (2.10) over $(t,t+1)$ and once more employing (2.12).

Now by a further testing procedure, we can turn the above information into the
estimate of $\|\nabla u(\cdot,t)\|_{L^{2}(\Omega)}$, particularly its decay in
the case of $r=0$, on the basis of an interpolation argument, which is
inspired by an approach illustrated in section 3.2 of [36].

######  Lemma 2.6.

For the global classical solution $(n,w,u,P)$ of (2.1), we have

i) if $r>0$, then there exists $\mu_{1}:=\mu_{1}(\Omega,r)>0$ such that for
all $\mu>\mu_{1}$,

$\displaystyle\displaystyle\lim\sup_{t\rightarrow\infty}\|\nabla
u(\cdot,t)\|_{L^{2}(\Omega)}\leq\frac{1}{17K_{1}|\Omega|}$ (2.13)

ii) if $r\leq 0$, then for any $\mu>0$,

$\displaystyle\displaystyle\lim_{t\rightarrow\infty}\|\nabla
u(\cdot,t)\|_{L^{2}(\Omega)}=0.$ (2.14)

Proof. Applying the Helmholtz projector $\mathcal{P}$ to the third equation in
(2.1), multiplying the resulting identity
$u_{t}+Au=-\mathcal{P}[(u\cdot\nabla)u]+\mathcal{P}[n\nabla\phi]$ by $Au$, and
using the Gagliardo–Nirenberg inequality, we can find $C_{1}>0$ such that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
u|^{2}+\int_{\Omega}|Au|^{2}$
$\displaystyle=-\int_{\Omega}\mathcal{P}[(u\cdot\nabla)u]\cdot
Au+\int_{\Omega}\mathcal{P}[n\nabla\phi]\cdot Au$
$\displaystyle\leq\frac{1}{2}\int_{\Omega}|Au|^{2}+\int_{\Omega}|(u\cdot\nabla)u|^{2}+\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\int_{\Omega}n^{2}$
$\displaystyle\leq\frac{1}{2}\int_{\Omega}|Au|^{2}+\|u\|^{2}_{L^{\infty}(\Omega)}\|\nabla
u\|^{2}_{L^{2}(\Omega)}+\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\int_{\Omega}n^{2}$
$\displaystyle\leq\frac{1}{2}\int_{\Omega}|Au|^{2}+C_{1}\|Au\|_{L^{2}(\Omega)}\|u\|_{L^{2}(\Omega)}\|\nabla
u\|^{2}_{L^{2}(\Omega)}+\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\int_{\Omega}n^{2}$
$\displaystyle\leq\int_{\Omega}|Au|^{2}+\frac{C^{2}_{1}}{2}\|u\|^{2}_{L^{2}(\Omega)}\|\nabla
u\|^{4}_{L^{2}(\Omega)}+\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\int_{\Omega}n^{2},$

which entails $y(t):=\int_{\Omega}|\nabla u(\cdot,t)|^{2}$ satisfies

$\displaystyle y^{\prime}(t)\leq h_{1}(t)y(t)+h_{2}(t)~{}~{}~{}~{}~{}\hbox{for
all}~{}~{}t>0$ (2.15)

with $h_{1}(t)=C^{2}_{1}\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}\|\nabla
u(\cdot,t)\|^{2}_{L^{2}(\Omega)}$ and
$h_{2}(t)=2\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\|n(\cdot,t)\|^{2}_{L^{2}(\Omega)}$.

i) In order to prepare the integration of (2.15), we may use Lemma 2.5 i) to
find some $t_{0}>0$ such that

$\displaystyle\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}\leq
C_{2}:=\frac{3(1+r)|\Omega|}{\mu}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{r}{\mu}$

and

$\displaystyle\int^{t}_{t-1}\|\nabla u(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds\leq
2C_{2}$

for all $t>t_{0}+1$.

Hence for any $t>t_{0}+1$, we can find $t_{*}=t_{*}(t)\in[t-1,t)$ such that

$\displaystyle\|\nabla u(\cdot,t_{*})\|^{2}_{L^{2}(\Omega)}\leq 2C_{2},$
(2.16)

and then integrating (2.15) over $(t_{*},t)$ yields

$\displaystyle y(t)$ $\displaystyle\leq
y(t_{*})e^{\int^{t}_{t_{*}}h_{1}(\sigma)d\sigma}+\int^{t}_{t_{*}}e^{\int^{t}_{s}h_{1}(\sigma)d\sigma}h_{2}(s)ds$
$\displaystyle\leq(2+C_{p})C_{2}e^{2C_{1}^{2}C_{2}^{2}}$

and thereby verifies (2.13).

ii) For any $t_{0}>1$ and $t>t_{0}+2$, we use Lemma 2.5 ii) to pick
$t_{*}=t_{*}(t)\in[t-1,t)$ fulfilling

$\displaystyle\|\nabla u(\cdot,t_{*})\|^{2}_{L^{2}(\Omega)}=$
$\displaystyle\int^{t}_{t-1}\|\nabla u(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds$
$\displaystyle\leq$
$\displaystyle\|u(\cdot,t_{0})\|^{2}_{L^{2}(\Omega)}e^{-\frac{C_{p}}{2}(t-1-t_{0})}+\frac{4|\Omega|}{\mu^{2}}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{1}{t_{0}+\gamma},$

as well as

$\displaystyle\int^{t}_{t-1}h_{1}(\sigma)d\sigma$ $\displaystyle\leq
C^{2}_{1}\max_{t-1\leq s\leq
t}\|u(\cdot,s)\|^{2}_{L^{2}(\Omega)}\int^{t}_{t-1}\|\nabla
u(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds$ $\displaystyle\leq
C^{2}_{1}(\|u(\cdot,t_{0})\|^{2}_{L^{2}(\Omega)}e^{-\frac{C_{p}}{2}(t-1-t_{0})}+\frac{4|\Omega|}{\mu^{2}}\frac{\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}}{C_{p}(1-e^{-\frac{C_{p}}{2}})}\frac{1}{t_{0}+\gamma})^{2}.$

In addition, by (2.12) we also have

$\displaystyle\int^{t}_{t-1}h_{2}(\sigma)d\sigma=2\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\int^{t}_{t-1}\|n(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds\leq
2\|\nabla\phi\|^{2}_{L^{\infty}(\Omega)}\frac{|\Omega|}{\mu^{2}}\frac{1}{t-1+\gamma}.$

Therefore combining the above inequalities, (2.15) implies that

$\displaystyle y(t)$ $\displaystyle\leq
y(t_{*})e^{\int^{t}_{t-1}h_{1}(\sigma)d\sigma}+e^{\int^{t}_{t-1}h_{1}(\sigma)d\sigma}\int^{t}_{t-1}h_{2}(s)ds$

and thus (2.14) holds readily.

##  3 Global boundedness of solutions

In this section, we show that the classical solution of the problem (2.1) is
globally bounded in the cases of $r>0$ and $r\leq 0$, respectively.

###  3.1 The case $r>0$

In this subsection, we derive the global boundedness of solutions to (2.1)
whenever $\mu$ is suitably large compared with $r$. As in [48], the main idea
is to examine the behavior of the functional

$\mathcal{F}(n,w):=\int_{\Omega}H(n)+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla
w|^{2}$ (3.1)

where $H(s):=s\ln\frac{\mu s}{er}+\frac{r}{\mu}$, along trajectories of the
boundary value problem (2.1).

The following elementary property of $H(n)$ will be used in the sequel.

######  Lemma 3.1.

For all nonnegative function $n\in C(\bar{\Omega})$, $H(n)\geq 0$.

Proof. It is easy to verify that $H(\frac{r}{\mu})=0$,
$H^{\prime}(\frac{r}{\mu})=0$ and $H^{\prime\prime}(s)=\frac{1}{s}\geq 0$,
which implies $H(n)\geq 0$ for all $n\geq 0$.

Now we can describe the evolution of $\mathcal{F}(n,w)$ along the trajectories
of (2.1) by the standard testing procedure.

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth bounded domain and $(n,w,u)$ be
the global classical solution of (2.1) with $r>0,\mu>0$. Then there exists
$t_{*}>0$ such that

$\frac{d}{dt}{\mathcal{F}(n,w)}\leq 0~{}~{}\hbox{for all}~{}~{}t\geq t_{*}$
(3.2)

whenever
$\mu>\mu_{2}(\Omega,\chi,r):=\max\\{\mu_{1},\frac{K_{1}(36+16\chi)|\Omega|}{\chi}r\\}$.

Proof. Multiplying the first equation in (2.1) by $H(n)$ and integrating by
parts, we get

$\begin{array}[]{rl}\displaystyle\frac{d}{dt}\int_{\Omega}H(n)=&\displaystyle\int_{\Omega}H^{\prime}(n)(\triangle
n+\chi\nabla\cdotp(n\nabla w)+rn-\mu n^{2}-u\cdot\nabla n)\\\
=&-\displaystyle\int_{\Omega}H^{\prime\prime}(n)(|\nabla n|^{2}+\chi n\nabla
n\cdot\nabla w)+\displaystyle\int_{\Omega}H^{\prime}(n)(rn-\mu
n^{2})\\\\[5.69054pt] =&-\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}-\chi\displaystyle\int_{\Omega}\nabla n\cdot\nabla
w+\displaystyle\displaystyle\int_{\Omega}(\ln n-\ln\frac{r}{\mu})(rn-\mu
n^{2})\\\\[5.69054pt]
\leq&-\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}-\chi\displaystyle\int_{\Omega}\nabla n\cdot\nabla w\\\ \end{array}$
(3.3)

due to $(\ln n-\ln\frac{r}{\mu})(rn-\mu n^{2})\leq 0$, $\nabla\cdot u=0$ and
$u=0$ on $\partial\Omega$.

On the other hand, testing the second equation in (2.1) by $-\triangle w$,
using $\nabla\cdot u=0$ and $u=0$ on $\partial\Omega$ again, we can obtain

$\begin{array}[]{rl}\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
w|^{2}+\int_{\Omega}|\triangle w|^{2}=&\displaystyle\int_{\Omega}|\nabla
w|^{2}\triangle w+\displaystyle\int_{\Omega}\nabla n\cdot\nabla
w+\int_{\Omega}(u\cdot\nabla w)\triangle w\\\\[5.69054pt]
\leq&\displaystyle\frac{1}{2}\int_{\Omega}|\triangle
w|^{2}+\displaystyle\frac{1}{2}\int_{\Omega}|\nabla
w|^{4}+\displaystyle\int_{\Omega}\nabla n\cdot\nabla
w+\int_{\Omega}(u\cdot\nabla w)\triangle w\\\\[5.69054pt]
=&\displaystyle\frac{1}{2}\int_{\Omega}|\triangle
w|^{2}+\displaystyle\frac{1}{2}\int_{\Omega}|\nabla
w|^{4}+\displaystyle\int_{\Omega}\nabla n\cdot\nabla w-\int_{\Omega}\nabla
w\cdot(\nabla u\cdot\nabla w).\end{array}$

Furthermore, by Lemma 2.2 i) and the Cauchy–Schwarz inequality, we get

$\begin{array}[]{rl}\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
w|^{2}+\displaystyle\frac{1}{2}\int_{\Omega}|\triangle
w|^{2}\leq&\displaystyle\frac{K_{1}}{2}\|\nabla
w\|^{2}_{L^{2}(\Omega)}\int_{\Omega}|\triangle
w|^{2}+\displaystyle\int_{\Omega}\nabla n\cdot\nabla
w+\displaystyle\int_{\Omega}|\nabla u||\nabla w|^{2}\\\\[5.69054pt]
\leq&(\displaystyle\frac{K_{1}}{2}\|\nabla
w\|^{2}_{L^{2}(\Omega)}+K_{1}|\Omega|^{\frac{1}{2}}\|\nabla
u\|_{L^{2}(\Omega)})\int_{\Omega}|\triangle
w|^{2}+\displaystyle\int_{\Omega}\nabla n\cdot\nabla w\par\end{array}$

and thus

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
w|^{2}+\displaystyle\frac{1}{2}(1-K_{1}\|\nabla
w\|^{2}_{L^{2}(\Omega)}-2K_{1}|\Omega|^{\frac{1}{2}}\|\nabla
u\|_{L^{2}(\Omega)})\int_{\Omega}|\triangle
w|^{2}\leq\displaystyle\int_{\Omega}\nabla n\cdot\nabla w.$ (3.4)

Since $2{\mathcal{F}(n,w)}\geq\chi\|\nabla w\|^{2}_{L^{2}(\Omega)}$ due to
$H(n)\geq 0$, combining (3.4) with (3.3) yields

$\frac{d}{dt}{\mathcal{F}(n,w)}+\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+(\displaystyle\frac{\chi}{2}-K_{1}{\mathcal{F}(n,w)}-2\chi
K_{1}|\Omega|^{\frac{1}{2}}\|\nabla
u\|_{L^{2}(\Omega)})\int_{\Omega}|\triangle w|^{2}\leq
0~{}~{}\hbox{for}~{}t>0.$ (3.5)

On the other hand, from (2.13), it is possible to pick some $t_{0}>0$ such
that

$16K_{1}|\Omega|^{\frac{1}{2}}\|\nabla
u(\cdot,t)\|_{L^{2}(\Omega)}<1~{}~{}~{}\hbox{for all }~{}t>t_{0}$

whenever $\mu>\mu_{1}$, and thereby

$\frac{d}{dt}{\mathcal{F}(n,w)}+\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+(\displaystyle\frac{3\chi}{8}-K_{1}\mathcal{F}(n,w))\int_{\Omega}|\triangle
w|^{2}\leq 0~{}~{}\hbox{for }~{}t>t_{0}.$ (3.6)

In what follows, we shall show that there exists $t_{*}>t_{0}$ such that
$4K_{1}\mathcal{F}(n,w)(t_{*})<\chi$.

Firstly by Lemma 2.3(i), there exists $t_{1}>t_{0}$ such that for all
$t>t_{1}$

$\|n(\cdot,t)\|_{L^{1}(\Omega)}\leq\frac{3|\Omega|r}{2\mu},$ (3.7)

which along with Lemma 2.3(iii) yields

$\begin{array}[]{rl}\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}|\nabla
w|^{2}\leq&\displaystyle\int_{\Omega}w(\cdot,t_{1})+\int^{t_{2}}_{t_{1}}\|n(\cdot,s)\|_{L^{1}(\Omega)}ds\\\
\leq&\displaystyle\int_{\Omega}w_{0}(x)+\int^{t_{1}}_{0}\|n(\cdot,s)\|_{L^{1}(\Omega)}ds+\frac{3|\Omega|r}{2\mu}(t_{2}-t_{1}).\end{array}$

Similarly invoking Lemma 2.3(i) and (ii), we find that

$\displaystyle\int^{t_{2}}_{t_{1}}\|n(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds\leq\frac{3|\Omega|}{2}(\frac{r}{\mu})^{2}(t_{2}-t_{1})+\frac{1}{\mu}\|n(\cdot,t_{1})\|_{L^{1}(\Omega)}.$

Hence there exists $t^{*}>t_{1}$ suitably large such that whenever $t_{2}\geq
t^{*}$,

$\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}|\nabla
w|^{2}\leq\frac{2|\Omega|r}{\mu}(t_{2}-t_{1})$ (3.8)

and

$\displaystyle\int^{t_{2}}_{t_{1}}\|n(\cdot,s)\|^{2}_{L^{2}(\Omega)}ds\leq
2|\Omega|(\frac{r}{\mu})^{2}(t_{2}-t_{1}).$ (3.9)

Let

$\mathcal{S}_{1}:=\\{t\in[t_{1},t_{2}]|\int_{\Omega}|\nabla
w(\cdot,t)|^{2}\geq\frac{8|\Omega|r}{\mu}\\}$

and

$\mathcal{S}_{2}:=\\{t\in[t_{1},t_{2}]|\|n(\cdot,t)\|^{2}_{L^{2}(\Omega)}\geq
8|\Omega|(\frac{r}{\mu})^{2}\\}.$

Then

$|\mathcal{S}_{1}|\leq\frac{|t_{2}-t_{1}|}{4},~{}~{}~{}|\mathcal{S}_{2}|\leq\frac{|t_{2}-t_{1}|}{4}.$
(3.10)

In order to estimate the size of $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$, we
recall (3.8) to get

$\frac{8|\Omega|r}{\mu}|\mathcal{S}_{1}|\leq\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}|\nabla
w|^{2}\leq\frac{2|\Omega|r}{\mu}(t_{2}-t_{1})$

and thus $|\mathcal{S}_{1}|\leq\frac{|t_{2}-t_{1}|}{4}$ is valid. Similarly,
one can verify that $|\mathcal{S}_{2}|\leq\frac{|t_{2}-t_{1}|}{4}$.

As (3.10) warrants that

$|(t_{1},t_{2})\setminus(\mathcal{S}_{1}\cup\mathcal{S}_{2})|\geq\frac{|t_{2}-t_{1}|}{2},$

one can conclude that there exists $t_{*}\in(t_{1},t_{2})$ such that

$\displaystyle\|n(\cdot,t_{*})\|^{2}_{L^{2}(\Omega)}<8|\Omega|(\frac{r}{\mu})^{2}$
(3.11)

and

$\int_{\Omega}|\nabla w(\cdot,t_{*})|^{2}<\frac{8|\Omega|r}{\mu}.$ (3.12)

Applying
$\xi\ln\frac{\xi}{\sigma}\leq\eta\xi^{2}+\ln\frac{1}{\eta\sigma}\cdot\xi$ for
all $\xi>0,\eta>0,\sigma>0$ (see Lemma 5.5 of [48]), and combining (3.7) with
(3.11), we then arrive at

$\begin{array}[]{rl}\displaystyle\int_{\Omega}H(n)(\cdot,t_{*})\leq&\displaystyle\frac{\mu}{r}\int_{\Omega}n^{2}(\cdot,t_{*})-\int_{\Omega}n(\cdot,t_{*})+\frac{r}{\mu}|\Omega|\\\\[8.53581pt]
\leq&\displaystyle\frac{9|\Omega|r}{\mu}.\end{array}$

Thereupon from (3.12) and the definition of $\mathcal{F}(n,w)$, it follows
that

$\mathcal{F}(n,w)(t_{*})<(9+4\chi)|\Omega|\frac{r}{\mu},$

which entails that $4K_{1}\mathcal{F}(n,w)(t_{*})<\chi$ provided
$\mu>\frac{K_{1}(36+16\chi)|\Omega|r}{\chi}$.

As an immediate consequence of (3.5), we have

$\frac{d}{dt}{\mathcal{F}(n,w)}+\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+\displaystyle\frac{\chi}{8}\int_{\Omega}|\triangle w|^{2}\leq
0~{}~{}\hbox{for all}~{}t>t_{*}$ (3.13)

when $\mu>\mu_{2}(\Omega,\chi,r)$, and thus end the proof of this lemma.

Additionally from (3.13), one can also conclude that

######  Corollary 3.1.

Under the conditions of Lemma 3.2, we have

$\mathcal{F}(n,w)(t)+\displaystyle\displaystyle\int^{t}_{t_{*}}\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+\displaystyle\frac{\chi}{8}\int^{t}_{t_{*}}\int_{\Omega}|\triangle
w|^{2}\leq(9+4\chi)|\Omega|\frac{r}{\mu}~{}~{}~{}\hbox{for all}~{}t>t_{*}.$
(3.14)

Next by a further testing procedure, we can turn the above information into
the uniform-in-time boundedness of $\|n(\cdot,t)\|_{L^{2}(\Omega)}$ and
$\|\nabla w(\cdot,t)\|_{L^{4}(\Omega)}$ if $\mu$ is appropriately large
compared with $r$, which will serve as the foundation for the proof of the
global boundedness of $\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ and $\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}$.

######  Lemma 3.3.

Under the assumptions in Lemma 3.2, there exists $C>0$ such that

$\displaystyle\|n(\cdot,t)\|_{L^{2}(\Omega)}+\|\nabla
w(\cdot,t)\|_{L^{4}(\Omega)}\leq C~{}~{}\hbox{for all}~{}~{}t\geq t_{*}$
(3.15)

provided $\mu>\frac{182K_{2}|\Omega|r}{\chi^{2}}$.

Proof. Multiplying the first equation in (2.1) by $n$ and integrating the
result over $\Omega$, we get

$\begin{array}[]{rl}\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}n^{2}=&-\displaystyle\int_{\Omega}|\nabla
n|^{2}-\chi\int_{\Omega}n\nabla n\nabla
w+r\int_{\Omega}n^{2}-\mu\int_{\Omega}n^{3}\\\\[5.69054pt]
\leq&-\displaystyle\frac{1}{2}\displaystyle\int_{\Omega}|\nabla
n|^{2}+\frac{1}{2}\int_{\Omega}n^{2}|\nabla
w|^{2}+r\int_{\Omega}n^{2}-\mu\int_{\Omega}n^{3}.\end{array}$ (3.16)

On the other hand, by the second equation in (2.1) and the identity $\nabla
w\cdot\nabla\Delta w=\frac{1}{2}\Delta|\nabla w|^{2}-|D^{2}w|^{2}$, we obtain

$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\int_{\Omega}|\nabla w|^{4}\\\
=&\displaystyle 2\int_{\Omega}|\nabla w|^{2}\triangle|\nabla
w|^{2}-4\int_{\Omega}|\nabla w|^{2}|D^{2}w|^{2}-4\int_{\Omega}|\nabla
w|^{2}\nabla w\cdot\nabla|\nabla w|^{2}\\\\[5.69054pt]
&+4\displaystyle\int_{\Omega}|\nabla w|^{2}\nabla n\cdot\nabla
w-4\displaystyle\int_{\Omega}|\nabla w|^{2}\nabla w\cdot\nabla(u\cdot\nabla
w)\\\\[5.69054pt] =&-\displaystyle 2\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}-4\int_{\Omega}|\nabla w|^{2}|D^{2}w|^{2}-4\int_{\Omega}|\nabla
w|^{2}\nabla w\cdot\nabla|\nabla w|^{2}-4\displaystyle\int_{\Omega}n|\nabla
w|^{2}\triangle w\\\\[8.53581pt] &-4\displaystyle\int_{\Omega}n\nabla|\nabla
w|^{2}\cdot\nabla w+2\displaystyle\int_{\partial\Omega}|\nabla
w|^{2}\frac{\partial|\nabla
w|^{2}}{\partial\nu}-4\displaystyle\int_{\Omega}|\nabla w|^{2}\nabla
w\cdot(\nabla u\cdot\nabla w)\end{array}$ (3.17)

due to $\nabla\cdot u=0$ and $u=0$ on $\partial\Omega$.

According to

$\frac{\partial|\nabla w|^{2}}{\partial\nu}\leq c_{1}|\nabla
w|^{2}~{}~{}\hbox{on}~{}\partial\Omega~{}\hbox{for some}~{}c_{1}>0$

and

$\||\nabla w|^{2}\|_{L^{2}(\partial\Omega)}\leq\eta\|\nabla|\nabla
w|^{2}\|_{L^{2}(\Omega)}+c_{2}(\eta)\||\nabla
w|^{2}\|_{L^{1}(\Omega)}~{}\hbox{for any}~{}\eta\in(0,\frac{5}{4})$

(see Lemma 4.2 of [30] and Remark 52.9 in [31]), one can conclude that

$2\displaystyle\int_{\partial\Omega}|\nabla w|^{2}\frac{\partial|\nabla
w|^{2}}{\partial\nu}\leq\frac{1}{4}\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+c_{3}(\int_{\Omega}|\nabla w|^{2})^{2}$ (3.18)

for some $c_{3}>0$.

For the other integrals on the right-side of (3.17), we use the Young
inequality to estimate

$-4\int_{\Omega}|\nabla w|^{2}\nabla w\cdot\nabla|\nabla
w|^{2}\leq\displaystyle\frac{1}{3}\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+12\int_{\Omega}|\nabla w|^{6}$ (3.19)
$-4\displaystyle\int_{\Omega}n\nabla|\nabla w|^{2}\cdot\nabla
w\leq\displaystyle\frac{1}{3}\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+12\int_{\Omega}n^{2}|\nabla w|^{2}$ (3.20)

as well as

$\begin{array}[]{rl}-4\displaystyle\int_{\Omega}n|\nabla w|^{2}\triangle
w&\leq\displaystyle\frac{1}{6}\int_{\Omega}|\nabla w|^{2}|\triangle
w|^{2}+24\int_{\Omega}n^{2}|\nabla w|^{2}\\\\[5.69054pt]
&\leq\displaystyle\frac{1}{3}\int_{\Omega}|\nabla
w|^{2}|D^{2}w|^{2}+24\int_{\Omega}n^{2}|\nabla w|^{2}\end{array}$ (3.21)

due to $|\triangle w|^{2}\leq 2|D^{2}w|^{2}$ on $\Omega$.

Substituting (3.18)–(3.21) into (3.17), we readily get

$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\int_{\Omega}|\nabla
w|^{4}+\frac{13}{12}\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+\frac{11}{3}\int_{\Omega}|\nabla w|^{2}|D^{2}w|^{2}\\\\[8.53581pt]
\leq&\displaystyle 12\int_{\Omega}|\nabla w|^{6}+36\int_{\Omega}n^{2}|\nabla
w|^{2}+c_{3}(\int_{\Omega}|\nabla w|^{2})^{2}+4\int_{\Omega}|\nabla
w|^{4}|\nabla u|\end{array}$

and thus

$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\int_{\Omega}|\nabla
w|^{4}+2\int_{\Omega}|\nabla|\nabla w|^{2}|^{2}\\\ \leq&\displaystyle
12\int_{\Omega}|\nabla w|^{6}+36\int_{\Omega}n^{2}|\nabla
w|^{2}+c_{3}(\int_{\Omega}|\nabla w|^{2})^{2}+4\int_{\Omega}|\nabla
w|^{4}|\nabla u|\end{array}$ (3.22)

due to the fact $|\nabla|\nabla w|^{2}|^{2}\leq 4|\nabla
w|^{2}|D^{2}w|^{2}~{}\hbox{on}~{}\Omega$.

Therefore combining (3.16) with (3.22) leads to

$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}(\int_{\Omega}n^{2}+\int_{\Omega}|\nabla
w|^{4})+2\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+\displaystyle\int_{\Omega}|\nabla n|^{2}\\\\[8.53581pt]
\leq&\displaystyle 12\int_{\Omega}|\nabla w|^{6}+37\int_{\Omega}n^{2}|\nabla
w|^{2}+c_{3}(\int_{\Omega}|\nabla
w|^{2})^{2}+2r\int_{\Omega}n^{2}-2\mu\int_{\Omega}n^{3}+4\int_{\Omega}|\nabla
w|^{4}|\nabla u|\\\ \leq&\displaystyle 13\int_{\Omega}|\nabla
w|^{6}+37^{2}\int_{\Omega}n^{3}+c_{3}(\int_{\Omega}|\nabla
w|^{2})^{2}+2r\int_{\Omega}n^{2}-2\mu\int_{\Omega}n^{3}+4\int_{\Omega}|\nabla
w|^{4}|\nabla u|.\end{array}$ (3.23)

Furthermore by Lemma 2.2 (ii), we get $\|\varphi\|^{3}_{L^{3}}\leq
K_{2}\|\nabla\varphi\|^{2}_{L^{2}}\|\varphi\|_{L^{1}}+c_{4}\|\varphi\|^{3}_{L^{1}}$
and thus

$\int_{\Omega}|\nabla w|^{6}\leq K_{2}(\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2})\left(\int_{\Omega}|\nabla
w|^{2}\right)+c_{4}(\int_{\Omega}|\nabla w|^{2})^{3}.$

Upon inserting this into (3.23) and (3.14), we obtain

$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}(\int_{\Omega}n^{2}+\int_{\Omega}|\nabla
w|^{4})+(2-13K_{2}\int_{\Omega}|\nabla w|^{2})\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+\displaystyle\int_{\Omega}|\nabla
n|^{2}+\int_{\Omega}n^{2}+\int_{\Omega}|\nabla w|^{4}\\\\[8.53581pt]
\leq&\displaystyle
37^{2}\int_{\Omega}n^{3}+(2r+1)\int_{\Omega}n^{2}-2\mu\int_{\Omega}n^{3}+\displaystyle\int_{\Omega}|\nabla
w|^{4}+4\int_{\Omega}|\nabla w|^{4}|\nabla u|+c_{5},\end{array}$

which, along with

$\int_{\Omega}|\nabla w|^{4}\leq\frac{1}{7}\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+c_{6}$

and

$\begin{array}[]{ll}4\displaystyle\int_{\Omega}|\nabla w|^{4}|\nabla u|&\leq
4\||\nabla w|^{2}\|^{2}_{L^{6}(\Omega)}\|\nabla
u\|_{L^{\frac{3}{2}}(\Omega)}\\\
&\leq\displaystyle\frac{13}{56}\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+c_{7}\\\ \end{array}$

by the Gagliardo–Nirenberg inequality and (2.13), implies that

$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}(\int_{\Omega}n^{2}+\int_{\Omega}|\nabla
w|^{4})+(\frac{13}{8}-13K_{2}\int_{\Omega}|\nabla
w|^{2})\int_{\Omega}|\nabla|\nabla
w|^{2}|^{2}+\displaystyle\int_{\Omega}|\nabla
n|^{2}+\int_{\Omega}n^{2}+\int_{\Omega}|\nabla w|^{4}\\\\[8.53581pt]
\leq&\displaystyle
37^{2}\int_{\Omega}n^{3}+(2r+1)\int_{\Omega}n^{2}-2\mu\int_{\Omega}n^{3}+c_{8}.\end{array}$
(3.24)

Finally according to an extended variant ([3]), (3.7) and (3.14), one can
infer that

$\begin{array}[]{rl}37^{2}\displaystyle\int_{\Omega}n^{3}\leq&c_{9}\displaystyle\left(\int_{\Omega}|\nabla
n|^{2}\right)\left(\int_{\Omega}n|\ln
n|\right)+c_{9}(\int_{\Omega}n)^{3}+c_{9}\\\\[5.69054pt]
\leq&\displaystyle\frac{1}{2}\int_{\Omega}|\nabla n|^{2}+c_{10}.\end{array}$

Hence, in conjunction with (3.14), (3.23) yields $c_{11}>0$ such that for all
$t>t_{*}$

$\displaystyle\frac{d}{dt}\int_{\Omega}(n^{2}+|\nabla
w|^{4})+\displaystyle\int_{\Omega}(n^{2}+|\nabla
w|^{4})+(\frac{13}{8}-13K_{2}\int_{\Omega}|\nabla
w|^{2})\int_{\Omega}|\nabla|\nabla w|^{2}|^{2}\leq c_{11}.$ (3.25)

Now in view of (3.14), we can see that $\int_{\Omega}|\nabla
w|^{2}\leq\frac{r}{\mu}(\frac{18}{\chi}+\frac{8}{\chi^{2}})|\Omega|$ for all
$t>t_{*}$ and thus

$\frac{13}{7}-13K_{2}\int_{\Omega}|\nabla w|^{2}\geq 0$

provided that $\frac{r}{\mu}<\frac{\chi^{2}}{182K_{2}|\Omega|}$, which
guarantees that

$\displaystyle\frac{d}{dt}\int_{\Omega}(n^{2}+|\nabla
w|^{4})+\displaystyle\int_{\Omega}(n^{2}+|\nabla w|^{4})\leq c_{11}$ (3.26)

for all $t>t_{*}$ and thereby (3.15) is valid.

We are now ready to prove Theorem 1.1 in the case of $r>0$.

Proof of Theorem 1.1 in the case of $r>0$. From the above lemmas, it follows
that there exists $C>0$ such that

$\displaystyle\|n(\cdot,t)\|_{L^{2}(\Omega)}+\|\nabla
w(\cdot,t)\|_{L^{4}(\Omega)}+\|\nabla u(\cdot,t)\|_{L^{2}(\Omega)}\leq
C~{}~{}$ (3.27)

whenever
$\mu>\mu_{0}(\chi,\Omega,r):=\max\\{\mu_{2}(\chi,\Omega,r),\frac{182K_{2}|\Omega|r}{\chi^{2}}\\}$.
Thereupon, by the argument in e.g. Lemma 4.4 of [4], we can readily prove that
$\|n(\cdot,t)\|_{L^{\infty}(\Omega)},\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}$ and
$\|A^{\alpha}u(\cdot,t)\|_{L^{2}(\Omega)}$ with some
$\alpha\in(\frac{1}{2},1)$ are globally bounded; we refer the reader to the
proof of Lemma 4.4 in [4], Lemma 3.12 and Lemma 3.11 in [36] for the details.

Based on the global boundedness of solutions, we are able to derive the
convergence result claimed in Theorem 1.1, namely,

$\displaystyle\lim_{t\rightarrow\infty}\|n(\cdot,t)-\frac{r}{\mu}\|_{L^{\infty}(\Omega)}=0,$
(3.28) $~{}~{}\displaystyle\lim_{t\rightarrow\infty}\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}=0,$ (3.29)
$~{}~{}\displaystyle\lim_{t\rightarrow\infty}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}=0$
(3.30)

as well as

$~{}~{}\displaystyle\lim_{t\rightarrow\infty}\displaystyle\inf_{x\in\Omega}w(x,t)=\infty.$
(3.31)

In fact, due to

$\int^{\infty}_{t_{*}}\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+\int^{\infty}_{t_{*}}\int_{\Omega}|\triangle w|^{2}\leq C$

established in (3.14), we can show (3.29), (3.30) and

$\displaystyle\lim_{t\rightarrow\infty}\|n(\cdot,t)-\overline{n}(t)\|_{L^{\infty}(\Omega)}=0$
(3.32)

with $\overline{n}(t)=\frac{1}{|\Omega|}\int_{\Omega}n(\cdot,t)$ by the
arguments in Proposition 4.15 of [4]. Therefore it suffices to show that

$\displaystyle\lim_{t\rightarrow\infty}|\overline{n}(t)-\frac{r}{\mu}|=0.$
(3.33)

To this end, we adapt the idea of [27] and give the details of the proof for
the convenience of readers.

Integrating the first equation in (2.1) on the spatial variable over $\Omega$,
we obtain

$\overline{n}_{t}=\lambda\overline{n}-\frac{\mu}{|\Omega|}\int_{\Omega}n^{2}=\lambda\overline{n}-\mu\overline{n}^{2}-\frac{\mu}{|\Omega|}\int_{\Omega}(n-\overline{n})^{2}.$

Putting
$a(t):=\frac{\mu}{|\Omega|}\int_{\Omega}(n(\cdot,t)-\overline{n})^{2}$, the
above equation then becomes

$\overline{n}_{t}=\mu\overline{n}(\frac{\lambda}{\mu}-\overline{n})-a(t)$
(3.34)

Thereupon multiplying (3.34) by $\overline{n}-\frac{\lambda}{\mu}$, we get

$\frac{d}{dt}(\overline{n}-\frac{\lambda}{\mu})^{2}+2\mu\overline{n}(\overline{n}-\frac{\lambda}{\mu})^{2}=-2a(t)(\overline{n}-\frac{\lambda}{\mu})$

and then

$2\mu\int^{\infty}_{1}\overline{n}(\overline{n}-\frac{\lambda}{\mu})^{2}\leq(\overline{n}(1)-\frac{\lambda}{\mu})^{2}+2\sup_{t\geq
1}|\overline{n}(t)-\frac{\lambda}{\mu}|\int^{\infty}_{1}a(t).$ (3.35)

In addition, invoking the Poincaré–Wintinger inequality

$\displaystyle\int_{\Omega}|\varphi-\frac{1}{|\Omega|}\int_{\Omega}\varphi(y)dy|^{2}\leq
C_{p}\int_{\Omega}|\varphi|\int_{\Omega}\frac{|\nabla\varphi|^{2}}{|\varphi|}~{}~{}\hbox{
for all~{}}\varphi\in W^{1,2}(\Omega)$

for some $C_{p}>0$, one can find

$\displaystyle\int^{\infty}_{1}a(s)ds\leq C_{p}\displaystyle\sup_{t\geq
1}\|n(t)\|_{L^{1}(\Omega)}\displaystyle\int^{\infty}_{1}\int_{\Omega}\frac{|\nabla
n(s)|^{2}}{n(s)}ds\leq C\\\ $ (3.36)

due to (3.14) and Lemma 2.5(i). Hence combining (3.36) with (3.35) yields

$\int^{\infty}_{1}\overline{n}(\overline{n}-\frac{\lambda}{\mu})^{2}\leq C.$
(3.37)

On the other hand,

$\frac{d}{dt}\overline{n}(\overline{n}-\frac{\lambda}{\mu})^{2}=\overline{n}_{t}((\overline{n}-\frac{\lambda}{\mu})^{2}+2\overline{n}(\overline{n}-\frac{\lambda}{\mu})),$

which along with
$|\overline{n}_{t}|\leq\lambda\overline{n}+\frac{\mu}{|\Omega|}\int_{\Omega}n^{2}\leq
C$ implies that

$\left|\frac{d}{dt}\overline{n}(\overline{n}-\frac{\lambda}{\mu})^{2}\right|\leq
C.$ (3.38)

Therefore by Lemma 6.3 of [27], (3.38) and (3.37) show that

$\displaystyle\lim_{t\rightarrow\infty}\overline{n}(t)(\overline{n}(t)-\frac{r}{\mu})^{2}=0.$
(3.39)

From (3.32), it follows that there exists $t_{1}>t_{*}$ such that
$\|n(\cdot,t)-\overline{n}(t)\|_{L^{\infty}(\Omega)}\leq\frac{\lambda}{2\mu}$
for all $t>t_{1}$, and thus

$\begin{array}[]{rl}\overline{n}_{t}=&\lambda\displaystyle\overline{n}-\mu\overline{n}^{2}-\displaystyle\frac{\mu}{|\Omega|}\int_{\Omega}n(n-\overline{n})\\\
\geq&\mu\overline{n}(\displaystyle\frac{\lambda}{\mu}-\overline{n}-\sup_{t>t_{1}}\|n(\cdot,t)-\overline{n}(t)\|_{L^{\infty}(\Omega)})\\\
\geq&\mu\overline{n}(\displaystyle\frac{\lambda}{2\mu}-\overline{n}).\end{array}$
(3.40)

On the other hand, noticing that the solution $y(t)$ of the ODE

$y^{\prime}(t)=\mu\overline{y}(\displaystyle\frac{\lambda}{2\mu}-\overline{y}),~{}~{}y(t_{1})>0$

satisfies $\displaystyle\lim_{t\rightarrow\infty}y(t)=\frac{r}{2\mu},$ by the
comparison principle, (3.40) implies that there exists $t_{2}>t_{1}$ such that
for all $t\geq t_{2}$,

$\overline{n}(t)\geq\displaystyle\frac{\lambda}{4\mu}.$

This together with (3.39) yields (3.33).

Finally, in view of (3.28), one can find $t_{3}>1$ such that
$n(x,t)\geq\frac{r}{2\mu}$ for all $x\in\Omega$ and $t\geq t_{3}$, and thereby
$w(x,t)$ satisfies

$w_{t}\geq\triangle w-|\nabla w|^{2}+\frac{r}{2\mu}-u\cdot\nabla w$

for $t\geq t_{3}$. Hence if $y(t)$ denotes the solution of ODE:
$y^{\prime}(t)=\frac{r}{2\mu},~{}~{}y(t_{3})=\displaystyle\min_{x\in\Omega}w(\cdot,t_{3})$,
then

$\displaystyle w(x,t)\geq\frac{r}{2\mu}(t-t_{3})$ (3.41)

by means of a straightforward parabolic comparison which warrants that (3.31)
holds and thereby completes the proof.

###  3.2 The case $r\leq 0$

In this subsection, we show the global boundedness of solutions to (1.1),
(1.4), (1.5) in the case $r\leq 0,\mu>0$. As mentioned in the introduction,
due to the structure of (1.1) with $r\leq 0,\mu>0$, it is difficult to find a
decreasing energy functional compared with the situation when $r>0,\mu>0$
considered in the previous subsection or when $r=\mu=0$ considered in [48].
Indeed, the energy-type functional $\mathcal{F}(n,w)$ in (3.1) of [48]
decreases along a solution in $\Omega\times(t_{0},\infty)$ if
$\mathcal{F}(n(\cdot,t_{0}),w(\cdot,t_{0}))$ is suitably small, namely

$\frac{d}{dt}{\mathcal{F}(n,w)}\leq 0~{}~{}\hbox{for all}~{}~{}t\geq t_{0}.$

The main idea underlying our approach is to make use of the quadratic
degradation in the first equation of $\eqref{1.1}$ which should enforce some
suitable regularity properties. More precisely, on the basis of (2.2), we can
show that the quantity of form

$\mathcal{F}(n,w):=\int_{\Omega}n(\ln
n+a)dx+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla w|^{2}dx,$ (3.42)

with parameter $a>0$ determined below (see (3.49)), satisfies a certain of
differential inequality. Although unlike the case of $r>0$ in which it enjoys
the monotonicity property, $\mathcal{F}(n,w)$ also provides us the global
boundedness of $\int_{\Omega}n|\ln n|dx$ and $\int_{\Omega}|\nabla w|^{2}dx$.
This is encapsulated in the following lemma.

######  Lemma 3.4.

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth bounded domain and $(n,w,u)$ be
the global classical solution (2.1) with $r\leq 0,\mu>0$. Then there exists
$t_{*}>0$ such that for all $t>t_{*}$

$\int_{\Omega}|\nabla w(\cdot,t)|^{2}\leq\frac{1}{4K_{1}}$ (3.43)

as well as

$\int_{\Omega}n|\ln n|\leq C$ (3.44)

for some $C>0$.

Proof. We test the first equation in (2.1) against $\ln n+a+1$, and integrate
by parts to see that

$\displaystyle\displaystyle\frac{d}{dt}\int_{\Omega}n(\ln n+a)$
$\displaystyle\leq$
$\displaystyle-\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}-\chi\displaystyle\int_{\Omega}\nabla n\cdot\nabla
w+\displaystyle\displaystyle\int_{\Omega}(n(r-\mu n)-u\cdot\nabla n)(\ln
n+a+1)$ $\displaystyle\leq$
$\displaystyle-\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}-\chi\displaystyle\int_{\Omega}\nabla n\cdot\nabla
w+\displaystyle\displaystyle\int_{\Omega}n(r-\mu n)(\ln n+a).$ (3.45)

due to $r\leq 0$ and $\nabla\cdot u=0$.

On the other hand, recalling (3.4) and (2.14), it is possible to fix $t_{0}>0$
such that for all $t\geq t_{0}$, we have

$\begin{array}[]{ll}&\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla
w|^{2}+\displaystyle\frac{1}{4}\int_{\Omega}|\triangle
w|^{2}+\displaystyle\frac{1}{4}(\frac{3}{4}-2K_{1}\|\nabla
w\|^{2}_{L^{2}(\Omega)})\int_{\Omega}|\triangle w|^{2}\\\
\leq&\displaystyle\int_{\Omega}\nabla u\cdot\nabla w.\end{array}$ (3.46)

From Lemma 2.2(i), there exists a constant $K_{3}>0$ such that

$\displaystyle 8K_{3}\|\nabla w\|^{2}_{L^{2}(\Omega)}\leq\|\Delta
w\|^{2}_{L^{2}(\Omega)}.$ (3.47)

Hence combining (3.46) with (3.2), we get

$\displaystyle\displaystyle\frac{d}{dt}{\mathcal{F}(n,w)}+\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+\displaystyle\frac{\chi}{4}\int_{\Omega}|\triangle
w|^{2}+K_{3}\int_{\Omega}n(\ln n+a)$
$\displaystyle+\frac{\chi}{4}(\frac{3}{4}-2K_{1}\|\nabla
w\|^{2}_{L^{2}(\Omega)})\int_{\Omega}|\triangle w|^{2}$ $\displaystyle\leq$
$\displaystyle\displaystyle\int_{\Omega}n(K_{3}-\mu n)(\ln
n+a)+r\displaystyle\int_{\Omega}n(\ln n+a)~{}~{}\hbox{for }~{}t\geq t_{0}.$
(3.48)

Now for any fixed
$\varepsilon<\min\\{\frac{\chi}{24K_{1}},\frac{\chi}{42K_{2}}\\}$, we pick
$a>1$ sufficiently large such that

$\displaystyle
e^{-a}<\frac{K_{3}}{\mu},~{}~{}~{}(1-r)|\Omega|\displaystyle\max_{0<n\leq
e^{-a}}|n\ln n|<\varepsilon\displaystyle\min\\{K_{3},1\\},$ (3.49)

due to $\displaystyle\lim_{n\rightarrow 0}n\ln n=0$ and $n\ln n<0$ for all
$n\in(0,1)$, and thereby fix $t_{1}>\max\\{1,t_{0}\\}$ fulfilling

$\displaystyle~{}~{}\displaystyle\frac{a|\Omega|}{\mu(t_{1}+\gamma)}<\frac{\varepsilon}{4},\displaystyle\frac{|\Omega|}{\mu^{2}(t_{1}+\gamma)}<\frac{\varepsilon}{16},\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}+\frac{2\chi(\frac{|\Omega|}{\mu}+\|n_{0}\|_{L^{1}(\Omega)}+\|w_{0}\|_{L^{1}(\Omega)})}{t_{1}}<\displaystyle\frac{\varepsilon}{4}.$
(3.50)

Let $t_{2}=t_{1}+t_{1}^{2}$,

$\mathcal{S}_{1}\triangleq\\{t\in[t_{1},t_{2}]|\int_{\Omega}|\nabla
w(\cdot,t)|^{2}\geq\frac{\varepsilon}{2\chi}\\}$

and

$\mathcal{S}_{2}\triangleq\\{t\in[t_{1},t_{2}]|\|n(\cdot,t)\|^{2}_{L^{2}(\Omega)}\geq\frac{\varepsilon}{4}\\}.$

Then

$|\mathcal{S}_{1}|\leq\frac{|t_{2}-t_{1}|}{4},~{}~{}~{}|\mathcal{S}_{2}|\leq\frac{|t_{2}-t_{1}|}{4}.$
(3.51)

By Lemma 2.3(iii), (2.2) and the second equation in (2.1), we obtain that

$\displaystyle\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}|\nabla
w|^{2}\leq$
$\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}n+\int_{\Omega}w(\cdot,t_{1})$
$\displaystyle=$
$\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}n+\int_{\Omega}w_{0}+\int^{t_{1}}_{0}\int_{\Omega}n$
$\displaystyle\leq$
$\displaystyle\displaystyle\frac{|\Omega|}{\mu(t_{1}+\gamma)}(t_{2}-t_{1})+\int_{\Omega}w_{0}+t_{1}\int_{\Omega}n_{0}.$

Furthermore, by (3.50)

$\displaystyle\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}|\nabla
w|^{2}dxds\leq$
$\displaystyle\displaystyle(\frac{|\Omega|}{\mu(t_{1}+\gamma)}+\frac{t_{1}\|n_{0}\|_{L^{1}(\Omega)}+\|w_{0}\|_{L^{1}(\Omega)}}{t_{2}-t_{1}})(t_{2}-t_{1})$
$\displaystyle\leq$
$\displaystyle\frac{\frac{|\Omega|}{\mu}+\|n_{0}\|_{L^{1}(\Omega)}+\|w_{0}\|_{L^{1}(\Omega)}}{t_{1}}(t_{2}-t_{1})$
$\displaystyle<$ $\displaystyle\frac{\varepsilon}{8\chi}(t_{2}-t_{1}).$

On the other hand, by the definition of $\mathcal{S}_{1}$, we see that

$\frac{\varepsilon}{2\chi}|\mathcal{S}_{1}|\leq\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}|\nabla
w|^{2}.$

and thereby $|\mathcal{S}_{1}|\leq\frac{|t_{2}-t_{1}|}{4}$.

In addition, by (2.2) and (3.50), we get

$\displaystyle\int^{t_{2}}_{t_{1}}\int_{\Omega}n^{2}\leq\frac{1}{\mu}\int_{\Omega}n(\cdot,t_{1})\leq\displaystyle\frac{|\Omega|}{\mu^{2}(t_{1}+\gamma)}<\frac{\varepsilon}{16},$

which implies that $|\mathcal{S}_{2}|\leq\frac{|t_{2}-t_{1}|}{4}$.

Therefore from (3.51), it follows that
$|(t_{1},t_{2})\setminus(\mathcal{S}_{1}\cup\mathcal{S}_{2})|\geq\frac{|t_{2}-t_{1}|}{2},$
and thereby there exists $t_{*}\in(t_{1},t_{2})$ such that

$\displaystyle\|n(\cdot,t_{*})\|^{2}_{L^{2}(\Omega)}<\frac{\varepsilon}{4}$
(3.52)

and

$\int_{\Omega}|\nabla
w(\cdot,t_{*})|^{2}<\frac{\varepsilon}{2\chi}<\frac{1}{6K_{1}}.$ (3.53)

By (3.53), we can see that the set

$\mathbf{S}\triangleq\\{t\in(t_{*},\infty)|~{}K_{1}\int_{\Omega}|\nabla
w(\cdot,t)|^{2}<\frac{1}{4}\\}$

is not empty and hence $T_{S}=\sup{\mathbf{S}}$ is a well-defined element of
$(t_{*},\infty]$. In fact, we claim that $T_{S}=\infty$. To this end,
supposing on the contrary that $T_{S}<\infty$, we then have
$K_{1}\int_{\Omega}|\nabla w(\cdot,t)|^{2}<\frac{1}{4}$ for all
$t\in[t_{*},T_{S})$, but

$K_{1}\int_{\Omega}|\nabla w(\cdot,T_{S})|^{2}=\frac{1}{4}.$ (3.54)

Hence from (3.48) and (3.47), it follows that for all $t\in[t_{*},T_{S})$,

$\displaystyle\displaystyle\frac{d}{dt}{\mathcal{F}(n,w)}+\displaystyle\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n}+\displaystyle\frac{\chi}{4}\int_{\Omega}|\triangle
w|^{2}+K_{3}\int_{\Omega}n(\ln n+a)+\frac{K_{3}\chi}{2}\int_{\Omega}|\nabla
w|^{2}$ $\displaystyle\leq$
$\displaystyle\displaystyle\int_{\Omega}n(K_{3}-\mu n)(\ln
n+a)+r\int_{\Omega}n(\ln n+a)$ $\displaystyle\leq$
$\displaystyle\displaystyle\int_{e^{-a}<n\leq\frac{K_{3}}{\mu}}n(K_{3}-\mu
n)(\ln n+a)+\displaystyle r\int_{0<n\leq e^{-a}}n(\ln n+a)$
$\displaystyle\leq$ $\displaystyle
K_{3}\displaystyle\int_{e^{-a}<n\leq\frac{K_{3}}{\mu}}n(\ln n+a)+\displaystyle
r\int_{0<n\leq e^{-a}}n\ln n$ (3.55) $\displaystyle\leq$ $\displaystyle
aK_{3}\displaystyle\int_{\Omega}n+K_{3}\displaystyle\int_{e^{-a}<n\leq\frac{K_{3}}{\mu}}n\ln
n-r|\Omega|\displaystyle\max_{0<n\leq e^{-a}}|n\ln n|$ $\displaystyle\leq$
$\displaystyle
K_{3}(a+(\ln\frac{K_{3}}{\mu})_{+})\displaystyle\int_{\Omega}n+\varepsilon
K_{3}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})K_{3}|\Omega|}{\mu(t_{1}+\gamma)}+\varepsilon
K_{3},$

where we have made use of $t_{*}\geq t_{1}$, the decay estimate (2.2) and
(3.50), and thus

$\displaystyle\mathcal{F}(n,w)(T_{s})+\displaystyle\int^{T_{s}}_{t_{*}}\displaystyle
e^{-K_{3}(T_{s}-\sigma)}(\int_{\Omega}\frac{|\nabla
n|^{2}}{n}(\cdot,\sigma)+\displaystyle\frac{\chi}{4}\int_{\Omega}|\triangle
w(\cdot,\sigma)|^{2})d\sigma$ $\displaystyle\leq$
$\displaystyle\mathcal{F}(n,w)(t_{*})+\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}+\varepsilon,$

which implies that

$\displaystyle\displaystyle\frac{\chi}{2}\int_{\Omega}|\nabla
w(\cdot,T_{S})|^{2}\leq$
$\displaystyle\mathcal{F}(n,w)(t_{*})+\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}-\int_{\Omega}n(\ln
n+a)(\cdot,T_{S})+\varepsilon$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}n(\ln
n+a)(\cdot,t_{*})+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla
w|^{2}(\cdot,t_{*})+\varepsilon$
$\displaystyle+\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}-\int_{\Omega}n(\ln
n+a)(\cdot,T_{S})$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}(n^{2}+an)(\cdot,t_{*})+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla
w|^{2}(\cdot,t_{*})+\varepsilon$ (3.56)
$\displaystyle+\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}-\int_{\Omega}n(\ln
n+a)(\cdot,T_{S}),$

due to $n\geq\ln n$ for all $n>0$.

In addition, by (3.50), we see that

$\displaystyle\int_{\Omega}n(\ln n+a)(\cdot,T_{S})\geq$
$\displaystyle\int_{0<n\leq e^{-a}}n(\ln n+a)(\cdot,T_{S})$ (3.57)
$\displaystyle\geq$ $\displaystyle\int_{0<n\leq e^{-a}}n\ln n(\cdot,T_{S})$
$\displaystyle\geq$ $\displaystyle-|\Omega|\displaystyle\max_{0<n\leq
e^{-a}}|n\ln n|$ $\displaystyle\geq$ $\displaystyle-\varepsilon.$

Upon inserting (3.57) into (3.56), we see that

$\displaystyle\displaystyle\frac{\chi}{2}\int_{\Omega}|\nabla
w(\cdot,T_{S})|^{2}\leq$
$\displaystyle\int_{\Omega}(n^{2}+an)(\cdot,t_{*})+\frac{\chi}{2}\displaystyle\int_{\Omega}|\nabla
w|^{2}(\cdot,t_{*})$ (3.58)
$\displaystyle+\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}+2\varepsilon,$

which along with (3.52), (3.53), (2.2) and (3.50), establishes that

$\displaystyle\displaystyle\frac{\chi}{2}\int_{\Omega}|\nabla
w(\cdot,T_{S})|^{2}\leq$
$\displaystyle\displaystyle\frac{5\varepsilon}{2}+a\int_{\Omega}n(\cdot,t_{*})+\displaystyle\frac{(a+(\ln\frac{K_{3}}{\mu})_{+})|\Omega|}{\mu(t_{1}+\gamma)}$
$\displaystyle<$ $\displaystyle 3\varepsilon$ (3.59) $\displaystyle\leq$
$\displaystyle\frac{\chi}{8K_{1}}.$

This contradicts (3.54) and thereby $T_{S}=\infty$, which means that the
differential inequality (3.55) is actually valid for all $t>t_{*}$.

Now revisiting the proof of (3.2), upon integration in time over $(t_{*},t)$,
we have

$\displaystyle\frac{\chi}{2}\int_{\Omega}|\nabla w(\cdot,t)|^{2}\leq
3\varepsilon~{}~{}~{}\hbox{for all}~{}~{}t>t_{*}$

which implies that (3.43) is valid by the choice of $\varepsilon$, as well as

$\int_{\Omega}n\ln n(\cdot,t)\leq C_{1}~{}~{}~{}\hbox{for all}~{}~{}t>t_{*}$
(3.60)

for some $C_{1}>0$.

Since $\xi\ln\xi\geq-\frac{1}{e}$ for all $\xi>0$,

$\displaystyle\int_{\Omega}n|\ln n|(\cdot,t)$ $\displaystyle=\int_{\Omega}n\ln
n(\cdot,t)-2\int_{0<n<1}n\ln n(\cdot,t)$ $\displaystyle\leq\int_{\Omega}n\ln
n(\cdot,t)+\frac{2|\Omega|}{e},$

which along with (3.60) readily implies that (3.44) is actually valid with
$C=C_{1}+\frac{2|\Omega|}{e}$.

Furthermore, from (3.55), one can also conclude that:

######  Corollary 3.2.

Under the conditions of Lemma 3.4, we have

$\displaystyle\lim_{t\rightarrow\infty}\int^{t+1}_{t}\int_{\Omega}(\frac{|\nabla
n|^{2}}{n}+|\triangle
w|^{2})=0,\quad\displaystyle\lim_{t\rightarrow\infty}\int_{\Omega}|\nabla
w(\cdot,t)|^{2}=0.$ (3.61)

Proof. On the basis of the decay estimate (2.2) and revisiting the argument in
the proof of Lemma 3.4, one can conclude that for any fixed
$\varepsilon\in(0,\frac{\chi}{42K_{1}})$, there exists $t_{\varepsilon}>1$
such that

$\int_{\Omega}|\nabla
w(\cdot,t)|^{2}+\displaystyle\int^{t}_{t_{\varepsilon}}\displaystyle
e^{-K_{3}(t-\sigma)}(\int_{\Omega}\frac{|\nabla
n|^{2}}{n}(\cdot,\sigma)+\displaystyle\frac{\chi}{8}\int_{\Omega}|\triangle
w(\cdot,\sigma)|^{2})d\sigma\leq\varepsilon$

for all $t>t_{\varepsilon}$. Furthermore, it follows from the above inequality
that

$\displaystyle\int^{t}_{t-1}\displaystyle(\int_{\Omega}\frac{|\nabla
n|^{2}}{n}(\cdot,\sigma)+\displaystyle\frac{\chi}{8}\int_{\Omega}|\triangle
w(\cdot,\sigma)|^{2})d\sigma\leq\varepsilon e^{K_{3}}$

for any $t>t_{\varepsilon}+1$, which implies that (3.61) is indeed valid.

At this point, we can prove Theorem 1.1 in the case of $r\leq 0$.

Proof of Theorem 1.1 in the case $r\leq 0$. We can repeat the argument in the
proof of Theorem 1.1 in the case $r>0$. In fact, in view of (3.43) and (3.44),
(3.15) is also valid for $r\leq 0,\mu>0$, and thereby the global boundedness
of solutions can be proven. In addition, similar to the case of $r>0$, we can
show

$\displaystyle\lim_{t\rightarrow\infty}\|n(\cdot,t)\|_{L^{\infty}(\Omega)}=0,$
(3.62) $~{}~{}\displaystyle\lim_{t\rightarrow\infty}\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}=0$ (3.63)

as well as

$\displaystyle\lim_{t\rightarrow\infty}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}=0.$
(3.64)

For the sake of completeness we shall only recount the main steps and refer to
the mentioned sources for more details. Invoking standard parabolic regularity
theory (see the proofs of Lemma 4.5 and Lemma 4.9 of [48] for details), one
can see that there exist $\theta\in(0,1)$ and $\alpha\in(\frac{1}{2},1)$ and
$c_{1}>0$ such that for all $t>1$

$\|n\|_{C^{\theta,\frac{\theta}{2}}(\overline{\Omega}\times[t,t+1])}+\|\nabla
w(\cdot,t)\|_{C^{\theta}(\overline{\Omega})}+\|A^{\alpha}u(\cdot,t)\|_{L^{2}(\Omega)}\leq
c_{1}.$ (3.65)

If (3.62) were false, then there would be $c_{2}>0$,
$(t_{k})_{k\in\mathbb{N}}$ and $(x_{k})_{k\in\mathbb{N}}\subseteq\Omega$ such
that $t_{k}\rightarrow\infty$ as $k\rightarrow\infty$, and
$n(x_{k},t_{k})>c_{2}$ for all $k\in\mathbb{N}$, which, along with the uniform
continuity of $n$ in $\overline{\Omega}\times[t,t+1]$ as shown by (3.65),
entails that one can find $r>0$ such that $B(x_{k},r)\subseteq\Omega$ for all
$k\in\mathbb{N}$ and

$n(x,t_{k})>\frac{c_{2}}{2}~{}~{}\hbox{for all}~{}~{}x\in B(x_{k},r).$

This shows

$\int_{\Omega}n(\cdot,t_{k})\geq\int_{B(x_{k},r)}n(\cdot,t_{k})\geq\frac{c_{2}}{2}\pi
r^{2}$

which contradicts (2.2) and thus proves (3.62). Similarly, on the basis of
(3.61) and (3.65), (3.63) can be proved. Finally, (3.64) results from (2.14),
(3.65) and a simple interpolation, and thereby completes the proof.

##  4 Asymptotic profile of solutions

It is observed that in the case $r<0$, solutions to (1.1), (1.4), (1.5) enjoy
the exponential decay property due to the exponential decay of
$\|n(\cdot,t)\|_{L^{1}(\Omega)}$. Therefore the present paper focuses on the
asymptotic profile of (1.1), (1.4), (1.5) in the cases $r>0$ and $r=0$,
namely, we will give the proofs of Theorems 1.2 and 1.3 respectively.

###  4.1 The case $r>0$

Making use of the convergence properties of $(n,\frac{|\nabla c|}{c})$
asserted in Theorem 1.1, we apply $L^{p}-L^{q}$ estimates for the Neumann heat
semigroup $(e^{t\Delta})_{t>0}$ to show
$(n,c,u)\rightarrow(\frac{r}{\mu},0,0)$ in $L^{\infty}(\Omega)$ and
$\frac{|\nabla c|}{c}\rightarrow 0$ in $L^{p}(\Omega)$ at some exponential
rate as $t\rightarrow\infty$, respectively, whenever $\mu$ is suitably large
compared with $r$. To this end, we first make an observation which will be
used in the proof of the subsequent lemma:

######  Lemma 4.1.

For any $\alpha\in(0,\min\\{\lambda_{1},r\\})$, for
$I:=\int_{0}^{\infty}(1+\sigma^{-\frac{2}{3}}+\sigma^{-\frac{1}{2}})e^{-(\lambda_{1}-\alpha)\sigma}d\sigma$,
and $c_{i}>0$ (i=1,3) as given by Lemma 2.1, there exist
$\varepsilon_{1}>0,\varepsilon_{2}>0$ such that

$\displaystyle
4c_{1}I\varepsilon_{2}<1,~{}~{}4c_{1}|\Omega|^{\frac{1}{6}}I\varepsilon_{1}\leq\varepsilon_{2},~{}~{}8\chi
c_{3}I\varepsilon_{2}<1,$ (4.1) $\displaystyle 8\mu\varepsilon_{1}<r-\alpha.$
(4.2)

######  Lemma 4.2.

Let $(n,w,u)$ be the global bounded solution of (2.1). If $\mu>32\chi
c_{3}c_{1}|\Omega|^{\frac{1}{6}}I^{2}r$, one can find constants $C_{i}>0$,
$i=1,2,3$, $\alpha\in(0,\min\\{\lambda_{1},r\\})$ and $\beta<\alpha$ such that

$\displaystyle\|n(\cdot,t)-\frac{r}{\mu}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq C_{1}e^{-\alpha t},$ (4.3) $\displaystyle\|\nabla
w(\cdot,t)\|_{L^{6}(\Omega)}$ $\displaystyle\leq C_{2}e^{-\alpha t}$ (4.4)

as well as

$\displaystyle\|u(\cdot,t)\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq
C_{3}e^{-\beta t}$ (4.5)

for all $t\geq 1$.

Proof. Let $N(x,t)=n(x,t)-\frac{r}{\mu}$, $\varepsilon_{1}>0$ and
$\varepsilon_{2}>0$ be given by Lemma 4.1. Then from (3.28), (3.29) and
(3.30), there exists $t_{0}>1$ suitably large such that for $t\geq t_{0}$

$\displaystyle\|N(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\frac{\varepsilon_{1}}{8},~{}~{}~{}(c_{2}+1)\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\frac{\varepsilon_{2}}{8}$ (4.6)

and

$\displaystyle
8c_{1}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\int_{0}^{\infty}(1+\sigma^{-\frac{1}{2}})e^{-(\lambda_{1}-\alpha)\sigma}d\sigma\leq
1.$ (4.7)

Now we consider

$\displaystyle
T\\!\triangleq\\!\sup\\!\left\\{\\!\widetilde{T}\\!\in\\!(t_{0},\infty)\\!\left|\begin{array}[]{ll}\\!\|N(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\varepsilon_{1}e^{-\alpha(t-t_{0})}\quad\hbox{for
all }\,t\in[t_{0},\widetilde{T}),\\\ \\!\|\nabla
w(\cdot,t)\|_{L^{6}(\Omega)}\leq\varepsilon_{2}e^{-\alpha(t-t_{0})}\quad\hbox{for
all }\,t\in[t_{0},\widetilde{T}).\end{array}\right.\\!\right\\}$ (4.10)

By (4.6), $T$ is well-defined. In what follows, we shall demonstrate that
$T=\infty$.

To this end, we first invoke the variation-of-constants representation of $w$:

$\begin{array}[]{ll}w(\cdot,t)=&e^{(t-t_{0})\Delta}w(\cdot,t_{0})-\displaystyle\int_{t_{0}}^{t}e^{(t-s)\Delta}|\nabla
w(\cdot,s)|^{2}ds+\int_{t_{0}}^{t}e^{(t-s)\Delta}N(\cdot,s)ds\\\\[5.69054pt]
&\displaystyle-\int_{t_{0}}^{t}e^{(t-s)\Delta}(u\cdot\nabla
w)(\cdot,s)ds+\frac{r}{\mu}(t-t_{0}),\end{array}$ (4.11)

and use Lemma 2.1(i), (ii) to estimate

$\displaystyle\|\nabla w(\cdot,t)\|_{L^{6}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\displaystyle\|\nabla
e^{(t-t_{0})\Delta}w(\cdot,t_{0})\|_{L^{6}(\Omega)}+\displaystyle\int_{t_{0}}^{t}\|\nabla
e^{(t-s)\Delta}|\nabla w(\cdot,s)|^{2}\|_{L^{6}(\Omega)}ds$
$\displaystyle+\displaystyle\int_{t_{0}}^{t}\|\nabla
e^{(t-s)\Delta}N(\cdot,s)\|_{L^{6}(\Omega)}ds+\displaystyle\int_{t_{0}}^{t}\|\nabla
e^{(t-s)\Delta}(u\cdot\nabla w)(\cdot,s)\|_{L^{6}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle 2c_{2}e^{-\lambda_{1}(t-t_{0})}\|\nabla
w(\cdot,t_{0})\|_{L^{6}(\Omega)}+c_{1}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{2}{3}})e^{-\lambda_{1}(t-s)}\|\nabla
w(\cdot,s)\|^{2}_{L^{6}(\Omega)}ds$
$\displaystyle+c_{1}|\Omega|^{\frac{1}{6}}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-\lambda_{1}(t-s)}\|N(\cdot,s)\|_{L^{\infty}(\Omega)}ds$
$\displaystyle+c_{1}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-\lambda_{1}(t-s)}\|u(\cdot,s)\|_{L^{\infty}(\Omega)}\|\nabla
w(\cdot,s)\|_{L^{6}(\Omega)}ds$ $\displaystyle:=$ $\displaystyle
I_{1}+I_{2}+I_{3}$ (4.12)

for all $t_{0}<t<T$.

Now we estimate the terms $I_{i}$ $(i=1,2,3)$ respectively. Firstly, from
(4.6), we have $I_{1}\leq\frac{\varepsilon_{2}}{4}e^{-\lambda_{1}(t-t_{0})}$.
By the definition of $T$ and (4.1), we can see that

$\displaystyle\begin{array}[]{rl}I_{2}\leq&c_{1}\varepsilon^{2}_{2}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{2}{3}})e^{-\lambda_{1}(t-s)}e^{-2\alpha(s-t_{0})}ds\\\\[5.69054pt]
\leq&c_{1}\varepsilon^{2}_{2}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{2}{3}})e^{-\lambda_{1}(t-s)}e^{-\alpha(s-t_{0})}ds\\\\[5.69054pt]
\leq&c_{1}\varepsilon^{2}_{2}\displaystyle\int_{0}^{\infty}(1+\sigma^{-\frac{2}{3}})e^{-(\lambda_{1}-\alpha)\sigma}d\sigma\cdot
e^{-\alpha(t-t_{0})}\\\
\leq&\displaystyle\frac{\varepsilon_{2}}{4}e^{-\alpha(t-t_{0})}.\end{array}$

By the definition of $T$, (4.7) and (4.1) again, we also have

$\displaystyle\begin{array}[]{rl}I_{3}\leq&(c_{1}|\Omega|^{\frac{1}{6}}\varepsilon_{1}+c_{1}\displaystyle\sup_{t\geq
t_{0}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\varepsilon_{2})\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-\lambda_{1}(t-s)}e^{-\alpha(s-t_{0})}ds\\\\[5.69054pt]
=&(c_{1}|\Omega|^{\frac{1}{6}}\varepsilon_{1}+c_{1}\displaystyle\sup_{t\geq
t_{0}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\varepsilon_{2})\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-(\lambda_{1}-\alpha)(t-s)}e^{-\alpha(t-t_{0})}ds\\\\[5.69054pt]
\leq&(c_{1}|\Omega|^{\frac{1}{6}}\varepsilon_{1}+c_{1}\displaystyle\sup_{t\geq
t_{0}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\varepsilon_{2})\displaystyle\int_{0}^{\infty}(1+\sigma^{-\frac{1}{2}})e^{-(\lambda_{1}-\alpha)\sigma}d\sigma\cdot
e^{-\alpha(t-t_{0})}\\\
\leq&\displaystyle\frac{3\varepsilon_{2}}{8}e^{-\alpha(t-t_{0})}.\end{array}$

Substituting these estimates into (4.12), we get

$\displaystyle\|\nabla
w(\cdot,t)\|_{L^{6}(\Omega)}\leq\frac{7\varepsilon_{2}}{8}e^{-\alpha(t-t_{0})}<\varepsilon_{2}e^{-\alpha(t-t_{0})}\quad\hbox{for
all }\,t\in[t_{0},T).$ (4.13)

On the other hand, since

$N_{t}=\triangle N+\chi\nabla\cdotp(n\nabla w)-rN-\mu N^{2}-u\cdot\nabla N,$

the variation-of-constants representation of $N$ yields

$\begin{array}[]{ll}N(\cdot,t)=&\displaystyle
e^{(t-t_{0})(\Delta-r)}N(\cdot,t_{0})+\chi\int_{t_{0}}^{t}e^{(t-s)(\Delta-r)}\nabla\cdotp(n\nabla
w)(\cdot,s)ds-\mu\int_{t_{0}}^{t}e^{(t-s)(\Delta-r)}N^{2}(\cdot,s)ds\\\
&-\displaystyle\int_{t_{0}}^{t}e^{(t-s)(\Delta-r)}(u\cdot\nabla
N)(\cdot,s)ds.\end{array}$

Then by $\nabla\cdot u=0$ we can see that

$\displaystyle\begin{array}[]{rl}&\|N(\cdot,t)\|_{L^{\infty}(\Omega)}\\\
\leq&\|\displaystyle
e^{(t-t_{0})(\Delta-r)}N(\cdot,t_{0})\|_{L^{\infty}(\Omega)}+\displaystyle\mu\int_{t_{0}}^{t}\|e^{(t-s)(\Delta-r)}N^{2}(\cdot,s)\|_{L^{\infty}(\Omega)}ds\\\
&+\displaystyle\int_{t_{0}}^{t}\|e^{(t-s)(\Delta-r)}\nabla\cdot(uN)(\cdot,s)\|_{L^{\infty}(\Omega)}ds+\displaystyle\chi\int_{t_{0}}^{t}\|e^{(t-s)(\Delta-r)}\nabla\cdot(n\nabla
w)(\cdot,s)\|_{L^{\infty}(\Omega)}ds\\\ :=&J_{1}+J_{2}+J_{3}+J_{4}.\\\
\end{array}$

Here the maximum principle together with (4.6) ensures that

$J_{1}\leq
e^{-r(t-t_{0})}\|N(\cdot,t_{0})\|_{L^{\infty}(\Omega)}\leq\frac{\varepsilon_{1}}{8}e^{-\alpha(t-t_{0})}.$

By the definition of $T$ and comparison principle, we infer that

$\displaystyle\begin{array}[]{rl}J_{2}\leq&\displaystyle\mu\int_{t_{0}}^{t}e^{-r(t-s)}\|e^{(t-s)\Delta}N^{2}(\cdot,s)\|_{L^{\infty}(\Omega)}ds\\\\[5.69054pt]
\leq&\displaystyle\mu\int_{t_{0}}^{t}e^{-r(t-s)}\|N(\cdot,s)\|^{2}_{L^{\infty}(\Omega)}ds\\\\[5.69054pt]
\leq&\mu\varepsilon^{2}_{1}\displaystyle\int_{t_{0}}^{t}e^{-r(t-s)}e^{-2\alpha(s-t_{0})}ds\\\\[5.69054pt]
\leq&\mu\varepsilon^{2}_{1}\displaystyle\int_{t_{0}}^{t}e^{-(r-\alpha)(t-s)}ds\cdot
e^{-\alpha(t-t_{0})}\\\\[5.69054pt]
\leq&\displaystyle\frac{\mu\varepsilon_{1}^{2}}{r-\alpha}e^{-\alpha(t-t_{0})}\\\
\leq&\displaystyle\frac{\varepsilon_{1}}{8}e^{-\alpha(t-t_{0})}\end{array}$

due to (4.2) and $\alpha<r$. Similarly by (4.7), we have

$\displaystyle\begin{array}[]{rl}J_{3}\leq&c_{1}\displaystyle\sup_{t\geq
t_{0}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-(\lambda_{1}+r)(t-s)}\|N(\cdot,s)\|_{L^{\infty}(\Omega)}ds\\\\[5.69054pt]
\leq&c_{1}\displaystyle\sup_{t\geq
t_{0}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\varepsilon_{1}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-(\lambda_{1}+r)(t-s)}e^{-\alpha(s-t_{0})}ds\\\\[5.69054pt]
\leq&c_{1}\displaystyle\sup_{t\geq
t_{0}}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\varepsilon_{1}\int_{0}^{\infty}(1+\sigma^{-\frac{1}{2}})e^{-(\lambda_{1}-\alpha)\sigma}d\sigma\cdot
e^{-\alpha(t-t_{0})}\\\
\leq&\displaystyle\frac{\varepsilon_{1}}{8}e^{-\alpha(t-t_{0})}.\end{array}$

As for the term $J_{4}$, we recall (4.1), (4.10) and apply Lemma 2.1(iii) to
get

$\displaystyle\begin{array}[]{rl}J_{4}\leq&\chi
c_{3}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{2}{3}})e^{-(\lambda_{1}+r)(t-s)}\|(n\nabla
w)(\cdot,s)\|_{L^{6}(\Omega)}ds\\\\[5.69054pt] \leq&\chi
c_{3}\varepsilon_{2}\displaystyle\int_{t_{0}}^{t}(1+(t-s)^{-\frac{2}{3}})e^{-(\lambda_{1}+r)(t-s)}(\frac{r}{\mu}+\varepsilon_{1}e^{-\alpha(s-t_{0})})e^{-\alpha(s-t_{0})}ds\\\\[5.69054pt]
\leq&\chi
c_{3}\varepsilon_{2}(\displaystyle\frac{r}{\mu}+\varepsilon_{1})\displaystyle\int_{0}^{\infty}(1+\sigma^{-\frac{2}{3}})e^{-(\lambda_{1}+r-\alpha)\sigma}d\sigma\cdot
e^{-\alpha(t-t_{0})}\\\
\leq&\displaystyle\frac{\varepsilon_{1}}{8}e^{-\alpha(t-t_{0})}+\chi
c_{3}\displaystyle\frac{r}{\mu}I\varepsilon_{2}e^{-\alpha(t-t_{0})}\\\
\leq&\displaystyle\frac{\varepsilon_{1}}{4}e^{-\alpha(t-t_{0})}\end{array}$

provided that

$\displaystyle\chi
c_{3}\varepsilon_{2}\displaystyle\frac{r}{\mu}I<\displaystyle\frac{\varepsilon_{1}}{8}.$
(4.14)

Therefore, letting
$\varepsilon_{2}=4c_{1}|\Omega|^{\frac{1}{6}}I\varepsilon_{1}$ in Lemma 4.1,
(4.14) can be warranted provided $\mu>32\chi
c_{3}c_{1}|\Omega|^{\frac{1}{6}}I^{2}r$, and thereby

$\|N(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\frac{5\varepsilon_{1}}{8}e^{-\alpha(t-t_{0})}\quad\hbox{for
all }\,t\in[t_{0},T).$

This along with (4.13) readily shows that $T$ cannot be finite. In combination
with the decay property (4.3), a straightford interpolation argument can be
employed to prove (4.5).

Proof of Theorem 1.2. According to (3.41) and
$w=-\ln(\frac{c}{\|c_{0}\|_{L^{\infty}(\Omega)}})$, we have
$c(x,t)\leq\|c_{0}\|_{L^{\infty}(\Omega)}e^{-\frac{r}{2\mu}(t-t_{3})}$ for all
$t\geq t_{3}$. On the other hand, if
$\mu_{*}(\chi,\Omega,r):=\max\\{\mu_{0},32\chi
c_{3}c_{1}|\Omega|^{\frac{1}{6}}I^{2}r\\}$, then as an immediate consequence
of Theorem 1.1 and Lemma 4.3, $n(\cdot,t)\rightarrow\frac{r}{\mu}$ and
$\frac{|\nabla c|}{c}(\cdot,t)\rightarrow 0$ in $L^{\infty}(\Omega)$ and
$L^{6}(\Omega)$, respectively, at an exponential rate when
$\mu>\mu_{*}(\chi,\Omega,r)$. Moreover, with the help of the uniform
boundedness of $\|\frac{|\nabla c|}{c}(\cdot,t)\|_{L^{\infty}(\Omega)}$ with
respect to $t>0$, one can show that $\frac{|\nabla c|}{c}(\cdot,t)\rightarrow
0$ in $L^{p}(\Omega)$ for any $p>1$ exponentially by the interpolation
argument. The proof of this theorem is thus complete.

###  4.2 The case $r=0$

The proof of Theorem 1.3 proceeds on an alternative reasoning. To this end,
making use of the decay information on $|\nabla w|$ in $L^{\infty}(\Omega)$ in
(3.61) and the quadratic degradation in the $n-$equation, we first turn the
decay property of $\|n(\cdot,t)\|_{L^{1}(\Omega)}$ from (2.2) into an upper
bound estimate of $\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$.

######  Lemma 4.3.

Let $(n,w,u)$ be the global bounded solution of (2.1) obtained in Theorem 1.1
with $r=0,\mu>0$. Then one can find constant $C>0$ such that

$\|n(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\frac{C}{t+1}~{}~{}\hbox{for
all}~{}~{}t>0.$ (4.15)

Proof. According to the known smoothing properties of the Neumann heat
semigroup $(e^{\tau\Delta})_{t>0}$ on $\Omega\subset\mathbb{R}^{n}$ (see
[43]), one can pick $c_{1}>0$ and $c_{2}>0$ such that for all $0<\tau\leq 1$,

$\|e^{\tau\Delta}\varphi\|_{L^{\infty}(\Omega)}\leq
c_{1}\tau^{-\frac{n}{2}}\|\varphi\|_{L^{1}(\Omega)}~{}~{}\hbox{for
all}~{}~{}\varphi\in L^{1}(\Omega)$ (4.16)

and

$\|e^{\tau\Delta}\nabla\cdot\varphi\|_{L^{\infty}(\Omega)}\leq
c_{2}\tau^{-\frac{1}{2}-\frac{n}{2p}}\|\varphi\|_{L^{p}(\Omega)}~{}~{}\hbox{for
all}~{}~{}\varphi\in C^{1}(\Omega;\mathbb{R}^{n}).$ (4.17)

By (3.63) and (3.64), there exists $t_{0}>3$ such that

$24c_{2}(\chi\|\nabla
w(\cdot,t)\|_{L^{3}(\Omega)}+\|u(\cdot,t)\|_{L^{2}(\Omega)})\leq 1~{}~{}\hbox{
for all}~{}~{}t>t_{0}-1.$ (4.18)

Now in order to prove the lemma, it is sufficient to derive a bound,
independent of $T\in(t_{0},\infty)$, for
$M(T)\triangleq\displaystyle\sup_{t_{0}-1<t<T}\\{t\|n(\cdot,t)\|_{L^{\infty}(\Omega)}\\}$.

By the variation-of-constants representation of $n$, we have

$\begin{array}[]{ll}n(\cdot,t)&=e^{\Delta}n(\cdot,t-1)+\displaystyle\chi\\!\int_{t-1}^{t}e^{(t-s)\Delta}\nabla\cdot(n\nabla
w)(\cdot,s)ds-\int_{t-1}^{t}e^{(t-s)\Delta}(u\cdot\nabla n)(\cdot,s)ds\\\
&-\displaystyle\mu\int_{t-1}^{t}e^{(t-s)\Delta}n^{2}(\cdot,s)ds.\end{array}$
(4.19)

Since $e^{(t-s)\Delta}$ is nonnegative in $\Omega$ for all $0<s<t$ due to the
maximum principle, it follows from the nonnegativity of $n$ that for all
$t\in(t_{0},T)$

$\displaystyle\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\|e^{\Delta}n(\cdot,t-1)\|_{L^{\infty}(\Omega)}+\chi\\!\int_{t-1}^{t}\|e^{(t-s)\Delta}\nabla\cdot(n\nabla
w)(\cdot,s)\|_{L^{\infty}(\Omega)}ds+\int_{t-1}^{t}\|e^{(t-s)\Delta}(u\cdot\nabla
n)(\cdot,s)\|_{L^{\infty}(\Omega)}ds$

which along with (4.16), (4.17), (4.18) and (2.2) yields

$\displaystyle\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq$
$\displaystyle
c_{1}\|n(\cdot,t-1)\|_{L^{1}(\Omega)}+c_{2}\chi\\!\int_{t-1}^{t}(t-s)^{-\frac{5}{6}}\|(n\nabla
w)(\cdot,s)\|_{L^{3}(\Omega)}ds+c_{2}\\!\int_{t-1}^{t}(t-s)^{-\frac{5}{6}}\|(un)(\cdot,s)\|_{L^{3}(\Omega)}ds$
$\displaystyle\leq$
$\displaystyle\displaystyle\frac{c_{1}|\Omega|}{\mu(t-1+\gamma)}+\frac{6c_{2}}{t-1}(\chi\displaystyle\max_{t_{0}-1<s<T}\|\nabla
w(\cdot,s)\|_{L^{3}(\Omega)}+\|u(\cdot,t)\|_{L^{2}(\Omega)})\\!\cdot M(T)$
$\displaystyle\leq$
$\displaystyle\displaystyle\frac{c_{1}|\Omega|}{\mu(t-1+\gamma)}+\frac{1}{4(t-1)}M(T).$

Hence,

$M(T)\leq\displaystyle\frac{4c_{1}|\Omega|}{\mu}+2\displaystyle\sup_{t_{0}-1<s<t_{0}}\\{s\|n(\cdot,s)\|_{L^{\infty}(\Omega)}\\},$

which readily yields (4.15) since $T>t_{0}$ is arbitrary, and thus ends the
proof.

In light of Lemma 4.3, we can derive a pointwise estimate $c(x,t)$ from below.

######  Lemma 4.4.

Let $(n,w,u)$ be the global classical solution of (2.1) obtained in Theorem
1.1 with $r=0,\mu>0$. Then there exists $\kappa>0$ fulfilling

$\displaystyle
c(x,t)\geq\frac{\displaystyle\inf_{x\in\Omega}c_{0}(x)}{(t+1)^{\kappa}}.$
(4.20)

Proof. By the second equation of (2.1) and Lemma 4.3, we can see that

$w_{t}\leq\triangle w-|\nabla w|^{2}+\frac{c_{1}}{t+1}-u\cdot\nabla w$

with some $c_{1}>0$ for all $t>0$. Let $y\in C^{1}([0,\infty))$ denote the
solution of the initial-value problem $y^{\prime}(t)=\frac{c_{1}}{t+1}$,
$y(0)=\|w_{0}\|_{L^{\infty}(\Omega)}$, then from the comparison principle, we
infer that

$\displaystyle
w(x,t)\leq\|w_{0}\|_{L^{\infty}(\Omega)}+c_{1}\ln(t+1)~{}~{}\hbox{ for
all}~{}t>0,$ (4.21)

which along with $w=-\ln(\frac{c}{\|c_{0}\|_{L^{\infty}(\Omega)}})$, yields
(4.20) with $\kappa=c_{1}$.

Now utilizing the decay information on $|\nabla w|$ in $L^{\infty}(\Omega)$ in
(3.61) again, and thanks to the precise information on the decay of
$\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ in Lemma 4.3, we can obtain the desired
estimate for $\|n(\cdot,t)\|_{L^{\infty}(\Omega)}$ from below as well as the
upper estimate for $\|\nabla w(\cdot,t)\|_{L^{\infty}(\Omega)}$.

######  Lemma 4.5.

Let $(n,w,u)$ be the solution of (2.1) obtained in Theorem 1.1 with
$r=0,\mu>0$. Then one can find $C_{1}>0$ and $C_{2}>0$ fulfilling

$\displaystyle\|n(\cdot,t)\|_{L^{\infty}(\Omega)}\geq\frac{1}{|\Omega|}\|n(\cdot,t)\|_{L^{1}(\Omega)}\geq\frac{C_{1}}{t+1}~{}~{}~{}\hbox{
for all}~{}~{}t>0$ (4.22)

as well as

$\displaystyle\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\frac{C_{2}}{t+1}~{}~{}\hbox{for
all}~{}~{}t>0.$ (4.23)

Proof. We first adapt the method in Lemma 4.3 to derive the precise decay rate
of $\|\nabla w(\cdot,t)\|_{L^{\infty}(\Omega)}$. By (3.63) and (3.64), one can
choose some $t_{0}>2$ such that

$4c_{1}\int^{\infty}_{0}(1+\sigma^{-\frac{1}{2}})e^{-\lambda_{1}\sigma}d\sigma(\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}+\|u(\cdot,t)\|_{L^{\infty}(\Omega)})\leq
1~{}~{}\hbox{ for all}~{}~{}t>\frac{t_{0}}{2},$ (4.24)

and then let
$M(T)\triangleq\displaystyle\sup_{\frac{t_{0}}{2}<s<T}\\{s\|\nabla
w(\cdot,s)\|_{L^{\infty}(\Omega)}\\}$ for all $T>t_{0}$.

By the variation-of-constants representation of $w$, we have

$w(\cdot,t)=e^{\frac{t}{2}\Delta}w(\cdot,\frac{t}{2})-\\!\int_{\frac{t}{2}}^{t}e^{(t-s)\Delta}|\nabla
w|^{2}(\cdot,s)ds+\int_{\frac{t}{2}}^{t}e^{(t-s)\Delta}(n-u\cdot\nabla
w)(\cdot,s)ds$

for all $t_{0}<t<T$. We then show that

$\displaystyle\|\nabla w(\cdot,t)\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\|\nabla
e^{\frac{t}{2}\Delta}w(\cdot,\frac{t}{2})\|_{L^{\infty}(\Omega)}+\\!\int_{\frac{t}{2}}^{t}\|\nabla
e^{(t-s)\Delta}|\nabla
w|^{2}\|_{L^{\infty}(\Omega)}+\int_{\frac{t}{2}}^{t}\|\nabla
e^{(t-s)\Delta}n\|_{L^{\infty}(\Omega)}$
$\displaystyle+\int_{\frac{t}{2}}^{t}\|\nabla e^{(t-s)\Delta}(u\cdot\nabla
w)\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq$ $\displaystyle
c_{1}(1+t^{-\frac{1}{2}})e^{-\frac{\lambda_{1}t}{2}}\|w(\cdot,\frac{t}{2})\|_{L^{\infty}(\Omega)}+c_{1}\int_{\frac{t}{2}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-\lambda_{1}(t-s)}\|n(\cdot,s)\|_{L^{\infty}(\Omega)}$
$\displaystyle+c_{1}\int_{\frac{t}{2}}^{t}(1+(t-s)^{-\frac{1}{2}})e^{-\lambda_{1}(t-s)}\|\nabla
w(\cdot,s)\|_{L^{\infty}(\Omega)}(\|\nabla
w(\cdot,s)\|_{L^{\infty}(\Omega)}+\|u(\cdot,s)\|_{L^{\infty}(\Omega)})$
$\displaystyle\leq$ $\displaystyle
c_{1}(1+t^{-\frac{1}{2}})e^{-\frac{\lambda_{1}t}{2}}(\|w_{0}\|_{L^{\infty}(\Omega)}+c_{2}\ln(t+1))+\frac{2c_{1}c_{2}}{t}\int^{\infty}_{0}(1+\sigma^{-\frac{1}{2}})e^{-\lambda_{1}\sigma}d\sigma$
$\displaystyle+\frac{2c_{1}}{t}\int^{\infty}_{0}(1+\sigma^{-\frac{1}{2}})e^{-\lambda_{1}\sigma}d\sigma\displaystyle\sup_{t\geq\frac{t_{0}}{2}}(\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}+\|u(\cdot,t)\|_{L^{\infty}(\Omega)})\cdot
M(T)$ $\displaystyle\leq$ $\displaystyle
c_{1}(1+t^{-\frac{1}{2}})e^{-\frac{\lambda_{1}t}{2}}(\|w_{0}\|_{L^{\infty}(\Omega)}+c_{2}\ln(t+1))+\frac{2c_{1}c_{2}}{t}\int^{\infty}_{0}(1+\sigma^{-\frac{1}{2}})e^{-\lambda_{1}\sigma}d\sigma$
$\displaystyle+\frac{1}{2t}M(T)$

by using Lemma 2.1(i), (4.21), (4.15) and (4.24). This along with the
definition of $M(T)$ yields

$M(T)\leq 2\displaystyle\sup_{\frac{t_{0}}{2}<s<t_{0}}\\{s\|\nabla
w(\cdot,s)\|_{L^{\infty}(\Omega)}\\}+c_{3}$

with some constant $c_{3}>0$ as
$\displaystyle\lim_{t\rightarrow\infty}t\ln(t+1)e^{-\lambda_{1}t}=0$. Hence,
upon the definition of $M(T)$, we arrive at (4.23) with an evident choice of
$C_{2}$.

Continuing with the proof, we claim that there exists $c_{4}>0$ such that

$\displaystyle\|n(\cdot,t)\|_{L^{\infty}(\Omega)}\geq\frac{1}{|\Omega|}\|n(\cdot,t)\|_{L^{1}(\Omega)}\geq\frac{c_{4}}{t+1}~{}~{}~{}\hbox{
for all}~{}~{}t>0.$ (4.25)

Indeed, from the $n-$equation of (2.1) with $r=0$ and Young’s inequality, it
follows that

$\displaystyle\displaystyle\frac{d}{dt}\int_{\Omega}\ln n$
$\displaystyle=\displaystyle\int_{\Omega}\frac{|\nabla
n|^{2}}{n^{2}}+\chi\displaystyle\int_{\Omega}\frac{1}{n}\nabla\cdot(n\nabla
w)-\mu\displaystyle\displaystyle\int_{\Omega}n$
$\displaystyle\geq-\frac{\chi^{2}}{4}\displaystyle\int_{\Omega}|\nabla
w|^{2}-\mu\displaystyle\displaystyle\int_{\Omega}n.$

Inserting (2.2) and (4.23) into the above inequality yields

$\displaystyle\displaystyle\frac{d}{dt}\int_{\Omega}\ln
n\geq-\frac{\chi^{2}}{4}\frac{C_{2}^{2}|\Omega|}{(t+1)^{2}}-\displaystyle\frac{|\Omega|}{t+\gamma}$

and thus

$\displaystyle\displaystyle\int_{\Omega}\ln
n(\cdot,t)\geq-|\Omega|\ln(t+\gamma)-c_{5}~{}~{}~{}\hbox{ for all}~{}~{}t>1$
(4.26)

with some $c_{5}>0$. On the other hand, by the Jensen inequality, we have

$\displaystyle|\Omega|\ln(\displaystyle\int_{\Omega}n(\cdot,t))-|\Omega|\ln|\Omega|=|\Omega|\ln\\{\frac{1}{|\Omega|}\int_{\Omega}n(\cdot,t))\\}\geq\displaystyle\int_{\Omega}\ln
n(\cdot,t).$

This inequality together with (4.26) readily leads to (4.22).

With the above lemmas at hand, we can now complete the proof of Theorem 1.3.

Proof of Theorem 1.3. By $w=-\ln(\frac{c}{\|c_{0}\|_{L^{\infty}(\Omega)}})$,
Lemma 4.3 and Lemma 4.5, one can see that $(n,\frac{|\nabla
c|}{c})\longrightarrow(0,0)$ in $L^{\infty}(\Omega)$ algebraically as
$t\rightarrow\infty$. Hence it suffices to show the decay property of
$c(x,t)$. In view of the $w$-equation in (2.1), (4.25), (4.23) and
$\nabla\cdot u=0$, we can pick $c_{1}>0$, $c_{2}>0$ and $c_{3}>0$ such that

$\displaystyle\displaystyle\frac{d}{dt}\int_{\Omega}w$
$\displaystyle=\displaystyle\int_{\Omega}n-\displaystyle\int_{\Omega}|\nabla
w|^{2}-\int_{\Omega}u\cdot\nabla w$
$\displaystyle\geq\frac{c_{1}|\Omega|}{t+1}-\frac{c_{2}|\Omega|}{(t+1)^{2}},~{}$

and hence

$\displaystyle\int_{\Omega}w(\cdot,t)\geq c_{1}|\Omega|\ln(t+1)-c_{3},$

which entails that for any $t>0$ there exists $x_{0}(t)\in\Omega$ such that

$w(x_{0}(t),t)\geq c_{1}\ln(t+1)-\frac{c_{3}}{|\Omega|}.$

Since for each $\varphi\in W^{1,p}(\Omega)$ with $p>2$, there exists $c_{4}>0$
such that

$|\varphi(x)-\varphi(y)|\leq
c_{4}|x-y|^{1-\frac{2}{p}}\|\nabla\varphi\|_{L^{p}(\Omega)}~{}~{}\hbox{ for
all}~{}x,y\in\Omega,$

we therefore obtain from (4.23) that

$\displaystyle w(x,t)$ $\displaystyle\geq w(x_{0}(t),t)-|x-x_{0}(t)|\|\nabla
w(\cdot,t)\|_{L^{\infty}(\Omega)}$ (4.27) $\displaystyle\geq
c_{1}\ln(t+1)-\frac{c_{3}}{|\Omega|}-c_{4}\hbox{diam}(\Omega),$

and thereby

$c(x,t)\leq\displaystyle\frac{c_{5}}{(t+1)^{c_{1}}}~{}~{}\hbox{ for
}~{}x\in\Omega,t>0$

with some $c_{5}>0$. This together with (4.20) shows that $c(x,t)$ actually
converges to 0 in $L^{\infty}(\Omega)$ algebraically as $t\rightarrow\infty$,
and thus ends the proof of Theorem 1.3.

## Acknowledgment

Peter Y. H. Pang was partially supported by the NUS AcRF grant
(R-146-000-249-114). The work of Yifu Wang was supported by the NNSFC grant
(No. 12071030) and funded by the Beijing Key Laboratory on MCAACI. Jingxue Yin
acknowledges support of NNSFC Grant (No. 11771156), Guangdong Basic and
Applied Basic Research Foundation Grant (No. 2020B1515310013) and NSF of
Guangzhou Grant (No. 201804010391).

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