# Chemotaxis systems with singular sensitivity and logistic source:
Boundedness, persistence, absorbing set, and entire solutions

Halil Ibrahim Kurt and Wenxian Shen
Department of Mathematics and Statistics
Auburn University
Auburn University, AL 36849
U.S.A

###### Abstract

This paper deals with the following parabolic-elliptic chemotaxis system with
singular sensitivity and logistic source,

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla
v)+u(a(t,x)-b(t,x)u),&x\in\Omega,\cr 0=\Delta v-\mu v+\nu
u,&x\in\Omega,\cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial
n}=0,&x\in\partial\Omega,\end{cases}$ (0.1)

where $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $a(t,x)$ and
$b(t,x)$ are positive smooth functions, and $\chi$, $\mu$ and $\nu$ are
positive constants. In recent years, a lot of attention has been drawn to the
question of whether logistic kinetics prevents finite-time blow-up in various
chemotaxis models. In the very recent paper [25], we proved that for given
nonnegative initial function $0\not\equiv u_{0}\in C^{0}(\bar{\Omega})$ and
$s\in\mathbb{R}$, (0.1) has a unique globally defined classical solution
$(u(t,x;s,u_{0}),v(t,x;s,u_{0}))$ with $u(s,x;s,u_{0})=u_{0}(x)$, provided
that $a_{\inf}=\inf_{t\in\mathbb{R},x\in\Omega}a(t,x)$ is large relative to
$\chi$ and $u_{0}$ is not small.

In this paper, we further investigate qualitative properties of globally
defined positive solutions of (0.1) under the assumption that $a_{\inf}$ is
large relative to $\chi$ and $u_{0}$ is not small. Among others, we provide
some concrete estimates for $\int_{\Omega}u^{-p}$ and $\int_{\Omega}u^{q}$ for
some $p>0$ and $q>\max\\{2,N\\}$ and prove that any globally defined positive
solution is bounded above and below eventually by some positive constants
independent of its initial functions. We prove the existence of a
“rectangular” type bounded invariant set (in $L^{q}$) which eventually
attracts all the globally defined positive solutions. We also prove that (0.1)
has a positive entire classical solution $(u^{*}(t,x),v^{*}(t,x))$, which is
periodic in $t$ if $a(t,x)$ and $b(t,x)$ are periodic in $t$ and is
independent of $t$ if $a(t,x)$ and $b(t,x)$ are independent of $t$.

Key words. Parabolic-elliptic chemotaxis system, logistic source, singular
sensitivity, global boundedness, absorbing set, entire positive solution,
pointwise persistence, stationary positive solution, periodic positive
solution.

##  1 Introduction and Main Results

Chemotaxis systems, also known as Keller-Segel systems, have been widely
studied since the pioneering works [22, 23] by Keller and Segel at the
beginning of 1970s on the mathematical modeling of the aggregation process of
Dictyostelium discoideum. The current paper is devoted to the study of the
asymptotic dynamics of the following parabolic-elliptic chemotaxis system,

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla
v)+u(a(t,x)-b(t,x)u),\quad&x\in\Omega,\cr 0=\Delta v-\mu v+\nu
u,\quad&x\in\Omega,\cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial
n}=0,\quad&x\in\partial\Omega,\end{cases}$ (1.1)

where $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $a(t,x)$ and
$b(t,x)$ are nonnegative smooth functions, and $\chi,\mu$ and $\nu$ are
positive constants. Biologically, (1.1) describes the evolution of a
biological process in which cells (with density $u$) move towards higher
concentrations of a chemical substance with density $v$ produced by cells
themselves. In (1.1), the cross-diffusion term
$-\chi\nabla\cdot(\frac{u}{v}\nabla v)$ reflects the chemotactic movement and
$\frac{\chi}{v}$ is refereed to as chemotaxis sensitivity; the reaction term
$u(a(t,x)-b(t,x)u)$ represents the cell kinetic mechanism and is referred to
as logistic source; $\mu>0$ denotes the degradation rate of the chemical
substance, and $\nu>0$ is the rate at which the mobile species produces the
chemical substance. It is seen that the chemotaxis sensitivity
$\frac{\chi}{v}$ is singular near $v=0$, reflecting an inhibition of
chemotactic migration at high signal concentrations. Such a sensitivity was
first proposed in [24] due to the Weber–Fechner law of stimulus perception.
The time and space dependence of the logistic source reflects the
heterogeneity of the underlying environment of the chemotaxis system.

Since the pioneering works of Keller and Segel ([22, 23, 24]), considerable
efforts have been devoted to identifying the effects of the cross-diffusion
and the kinetic term on the blow-up or global boundedness of solutions of
(1.1). For example, consider (1.1) without logistic source (i.e.
$a(x,t)=b(x,t)\equiv 0$) and $\mu=\nu=1$. When $\Omega$ being a ball, it is
shown in [30] that the classical radially symmetric positive solutions are
global and bounded when $\chi>0$ and $N=2$, or $\chi<\frac{2}{N-2}$ and $N\geq
3$, and there exist radial blow-up solutions if $\chi>\frac{2N}{N-2}$ and
$N\geq 3$. Without the requirement for symmetry, Biler in [3] proved the
global existence of positive solutions when $\chi\leq 1$ and $N=2$, or
$\chi<\frac{2}{N}$ and $N\geq 2$. Fujie, Winkler, and Yokota in [13] proved
the boundedness of globally defined positive solutions when $\chi<\frac{2}{N}$
and $N\geq 2$. More recently, Fujie and Senba in [11] proved the global
existence and boundedness of classical positive solutions for the case of
$N=2$ for any $\chi>0$. The existence of finite-time blow-up is then
completely ruled out for any $\chi>0$ in the case $N=2$. In [4], global
existence of weak solutions is proved if $0<\chi<\frac{N}{N-2}$.

Consider (1.1) with $a(t,x),b(t,x)>0$. Central questions include whether the
logistic source prevents the occurrence of finite-time blow-up in (1.1) (i.e.
any positive solution exists globally); if so, whether the logistic source
prevents the occurrence of infinite-time blow-up in (1.1) (i.e. any globally
defined positive solution is bounded), and what is the long time behavior of
globally defined bounded positive solutions, etc. To recall the existing
results related to these central questions, we first make the following
standing assumption on $a(t,x)$ and $b(t,x)$:

(H) $a(t,x)$ and $b(t,x)$ are continuous in $x\in\bar{\Omega}$ uniformly with
respect to $t\in\mathbb{R}$,  Hölder continuous in $t\in\mathbb{R}$ with
exponent $0<\gamma_{0}<1$ uniformly with respect to $x\in\bar{\Omega}$, i.e,
there is $B_{1}>0$ such that

$|a(t,x)-a(s,x)|\leq B_{1}|t-s|^{\gamma_{0}},\quad|b(t,x)-b(s,x)|\leq
B_{1}|t-s|^{\gamma_{0}}\quad\forall\,t,s\in\mathbb{R},\,\,x\in\bar{\Omega},$

and there are positive constants $\alpha$, $B_{2}$ such that

$\alpha\leq a(t,x)\leq B_{2},\quad\alpha\leq b(t,x)\leq B_{2}.$

Put

$a_{\inf}=\inf_{x\in\bar{\Omega},t\in\mathbb{R}}a(t,x),\,\,b_{\inf}=\inf_{x\in\bar{\Omega},t\in\mathbb{R}}b(t,x),\,\,a_{\sup}=\sup_{x\in\bar{\Omega},t\in\mathbb{R}}a(t,x),\,\,b_{\sup}=\sup_{x\in\bar{\Omega},t\in\mathbb{R}}b(t,x).$

We consider the classical solutions $(u(t,x),v(t,x))$ of (1.1) with initial
functions $u_{0}(x)$ satisfying

$u_{0}\in C^{0}(\bar{\Omega}),\quad u_{0}\geq 0,\quad{\rm
and}\quad\int_{\Omega}u_{0}>0.$ (1.2)

######  Definition 1.1.

For given $s\in\mathbb{R}$ and $u_{0}$ satisfying (1.2), we say
$(u(t,x),v(t,x))$ is a classical solution of (1.1) on $(s,s+T)$ for some
$T\in(0,\infty]$ with initial condition $u(s,x)=u_{0}(x)$ if

$u(\cdot,\cdot)\in C([s,s+T)\times\bar{\Omega})\cap
C^{1,2}((s,s+T)\times\bar{\Omega}),\quad v(\cdot,\cdot)\in
C^{0,2}((s,s+T)\times\bar{\Omega}),$ $\lim_{t\to
s+}\|u(t,\cdot;s,u_{0})-u_{0}(\cdot)\|_{C^{0}(\bar{\Omega})}=0,$ (1.3)

and $(u(t,x),v(t,x))$ satisfies (1.1) for all $(t,x)\in(s,s+T)\times\Omega$.

Sometime, we may assume

$a_{\inf}>\begin{cases}\frac{\mu\chi^{2}}{4},&\text{if $0<\chi\leq 2,$}\\\
\mu(\chi-1),&\text{if $\chi>2$.}\\\ \end{cases}$ (1.4)

The following proposition is on the existence and uniqueness of the classical
solutions of (1.1) with given initial function $u_{0}$ satisfying (1.2) and
follows from the arguments in [12, Lemma 2.2].

######  Proposition 1.1.

(Local existence) For any $s\in\mathbb{R}$ and $u_{0}$ satisfying (1.2), there
is $T_{\max}(s,u_{0})\in(0,\infty]$ such that the system (1.1) possesses a
unique classical solution, denoted by $(u(t,x;s,u_{0})$, $v(t,x;s,u_{0}))$, on
$(s,T_{\max}(s,u_{0}))$ with initial condition $u(s,x;s,u_{0})=u_{0}(x)$.
Furthermore, if $T_{\max}(s,u_{0})<\infty,$ then

$\limsup_{t\nearrow
T_{\max}(s,u_{0})}\left\|u(t,\cdot;s,u_{0})\right\|_{C^{0}(\bar{\Omega})}=\infty\quad
or\quad\liminf_{t\nearrow
T_{\max}(s,u_{0})}\inf_{x\in\Omega}v(t,\cdot;s,u_{0})=0.$

Note that, by the assumption (H), it is not difficult to prove that
$\inf_{s\in\mathbb{R}}T_{\max}(s,u_{0})>0$.

We say that finite-time blow-up occurs in (1.1) if for some $s\in\mathbb{R}$
and $u_{0}$ satisfying (1.2), $T_{\max}(s,u_{0})<\infty$, and infinite-time
blow-up occurs if for some $s\in\mathbb{R}$ and $u_{0}$ satisfying (1.2),
$T_{\max}(s,u_{0})=\infty$ and

$\limsup_{t\to\infty}\left\|u(t,\cdot;s,u_{0})\right\|_{C^{0}(\bar{\Omega})}=\infty\quad
or\quad\liminf_{t\to\infty}\inf_{x\in\Omega}v(t,\cdot;s,u_{0})=0.$

Throughout the rest of this section, we assume that $u_{0}(x)$ satisfies (1.2)
and $(u(t,x),v(t,x)):=(u(t,x;s,u_{0}),v(t,x;s,u_{0}))$ is the unique classical
solution of (1.1) on $(s,T_{\max}(s,u_{0}))$ with the initial condition
$u(s,x;s,u_{0})=u_{0}(x)$. We now recall some existing results related to the
central questions mentioned in the above.

When $N=2$ and $a(t,x)\equiv a$, $b(t,x)\equiv b$ are positive constants, the
following have been proved:

  * (a)

For any $u_{0}$ satisfying (1.2) and $s\in\mathbb{R}$,
$T_{\max}(s,u_{0})=\infty$, that is, finite-time blow-up does not occur (see
[12, Theorem 1.1]).

  * (b)

If (1.4) holds, then for any $u_{0}$ satisfying (1.2) and $s\in\mathbb{R}$,
$\sup_{t\geq s}\|u(t,\cdot;s,u_{0})\|_{\infty}<\infty$, that is, infinite-time
blow-up does not occur under the assumption (1.4) ([12, Theorem 1.2]).

  * (c)

The constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ is
exponentially stable under the assumption (1.4) and some other assumptions
(see [6, Theorem 1]).

The authors of the current paper studied the global existence of classical
solutions of (1.1) for general $N\geq 1$ and time and space dependent
functions $a(t,x)$, $b(t,x)$ in [25]. To recall some results proved in [25],
we first state some additional conditions on initial data $u_{0}$ and on the
coefficients in (1.1). Here is an additional condition on the initial function
$u_{0}$:

$u_{0}\,\,\text{satisfies}\,\,\eqref{initial-cond-eq}\,\,\,{\rm
and}\,\,\,\exists\,\tau_{0}>0\,\,\,{\rm
s.t.}\,\,\,\int_{\Omega}u^{-1}(s+\tau_{0},x;s,u_{0})\leq\frac{b_{\sup}|\Omega|\max\\{1,\frac{1}{\chi}\\}}{a_{\inf}-a_{\chi,\mu}}\,\,\,\forall\,s\in\mathbb{R}.$
$None$

By [25, Proposition 1.3] (see also Lemma 3.2), for any $q\geq 3$ and $q-1\leq
k<2q-2,$ there exist positive constants $M(k,q)>0$ and $M^{*}(k,q)>0$ such
that for any given $s\in\mathbb{R}$ and $u_{0}$ satisfying (1.2),

$\int_{\Omega}\frac{|\nabla v(x,t;s,u_{0})|^{2q}}{v^{k+1}(x,t;s,u_{0})}\leq
M(k,q)\int_{\Omega}\frac{u^{q}(x,t;s,u_{0})}{v^{k-q+1}(x,t;s,u_{0})}+M^{*}(k,q)\int_{\Omega}v^{2q-k-1}(x,t;s,u_{0})$

for all $t\in(s,T_{\rm max}(s,u_{0}))$. The following is an additional
condition on the parameters:

$a_{\inf}>a_{\chi,\mu}+\frac{b_{\sup}|\Omega|(p_{N}-1)\big{(}C_{n}^{*}\big{)}^{\frac{1}{p_{N}+1}}\max\\{\chi,\chi^{2}\\}}{4b_{\inf}\delta_{0}},$
$None$

where $\delta_{0}$ is as in Lemma 2.3, $p_{N}=\max\\{2,N\\}$,
$C_{N}^{*}=M(p_{N},p_{N}+1)$, and

$a_{\chi,\mu}:=2\big{(}\chi+2-2\sqrt{\chi+1}\big{)}\mu.$

Note that if
$\int_{\Omega}u_{0}^{-1}(x)dx\leq\frac{b_{\sup}|\Omega|\max\\{1,\frac{1}{\chi}\\}}{a_{\inf}-a_{\chi,\mu}}$,
then $(1.2)^{\prime}$ holds. The condition $(1.2)^{\prime}$ indicates that
$u_{0}$ is not small, which prevents $v$ becomes too small as time evolutes
and is a natural assumption. Note also that $a_{\chi,\mu}$ in $(1.5)^{\prime}$
satisfies

$a_{\chi,\mu}\leq\begin{cases}\frac{\mu\chi^{2}}{2},\quad&{\rm
if}\,\,\,\,0<\chi\leq 2\cr 2\mu(\chi-1),\quad&{\rm
if}\,\,\,\,\chi>2.\end{cases}$

The condition $(1.5)^{\prime}$ indicates that $a(\cdot,\cdot)$ is large
relative to the chemotaxis sensitivity coefficient $\chi$, which also prevents
$v$ becomes too small as time evolutes, and is a natural condition. Assuming
that $(1.2)^{\prime}$ and $(1.5)^{\prime}$ hold, among others, the following
are proved in the recent paper [25]:

  * (i)

(Global existence) For any $s\in\mathbb{R}$,

$T_{\max}(s,u_{0})=\infty$ (1.5)

(see [25, Theorem 1.2(3)]).

  * (ii)

(Boundedness of $\int_{\Omega}u^{q}$) There is $q>N$ such that for any
$s\in\mathbb{R}$,

$\sup_{t\geq s}\int_{\Omega}u^{q}(t,x;s,u_{0})dx<\infty$ (1.6)

(see [25, Theorem 1.1(3)]).

  * (iii)

(Boundedness of $\int_{\Omega}u^{-p}$) There is $p>0$ such that for any
$s\in\mathbb{R}$,

$\limsup_{t\to\infty}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx<\infty$ (1.7)

(see [25, Lemma 3.4]).

  * (iv)

(Mass persistence) For any $s\in\mathbb{R}$,

$\inf_{t\geq s}\int_{\Omega}u(t,x;s,u_{0})dx>0,\quad\inf_{t\geq
s,x\in\Omega}v(t,x;s,u_{0})>0$ (1.8)

(see [25, Proposition 1.2(2)]).

  * (v)

(Boundedness of $\|u\|_{\infty}$) For any $s\in\mathbb{R}$,

$\sup_{t\geq s}\|u(t,\cdot;s,u_{0})\|_{\infty}<\infty$ (1.9)

(see [25, Theorem 1.2(3)]).

We remark that (1.9) implies (1.6). But (1.9) is proved in [25] by using (1.6)
and (1.8), and (1.8) is proved in [25] by using (1.7). Hence the boundedness
of $\int_{\Omega}u^{q}$ in (1.6) and the boundedness of $\int_{\Omega}u^{-p}$
in (1.7) play essential roles in the proofs of the main results in [25].

The results (a)-(c) and (i)-(v) provide some answers to the central questions
mentioned in the above. For example, results (i)-(v) imply that logistic
kinetics prevents the occurrence of finite-time blow-up provided that the
intrinsic growth rate function $a(t,x)$ is large relative to the chemotaxis
sensitivity and the initial condition $u_{0}$ is not small, which is an
interesting biological phenomenon. However, there are still many interesting
questions associated to those central questions. For example, whether the
ultimate upper bound of $\int_{\Omega}u^{q}$, $\int_{\Omega}u^{-p}$, and
$\|u\|_{\infty}$ in (ii), (iii), and (v), respectively, are independent of the
initial functions; whether the mass persistence in (iv) (i.e. $\int_{t\geq
s}\int_{\Omega}u(t,x;s,u_{0})dx>0$) can be replaced by the pointwise
persistence (i.e. $\liminf_{t-s\to\infty}\inf_{x\in\Omega}u(t,x;s,u_{0})>0$);
whether (1.1) has bounded positive entire solutions (i.e. bounded positive
solutions $(u(t,x),v(t,x))$ which are defined for all $t\in\mathbb{R}$);
whether (i)-(v) hold without the assumptions $(1.2)^{\prime}$ and
$(1.5)^{\prime}$.

In the current paper, we will further investigate those central questions
mentioned in the above. We state our main results in next subsection.

###  1.1 Main results and remarks

Assume that $(1.5)^{\prime}$ holds. For any $s\in\mathbb{R}$, $\tau\geq s$,
$u_{0}$ satisfying $(1.2)^{\prime}$, and $p>0$, let

$m^{*}(\tau,s,u_{0})={\rm
max}\Big{\\{}\int_{\Omega}u(\tau,x;s,u_{0})dx,\frac{a_{\sup}}{b_{\inf}}|\Omega|\Big{\\}}$

and

$\tilde{M}_{1}(p,\tau,s,u_{0})=pb_{\sup}|1-p|\Big{(}m^{*}(\tau,s,u_{0})-\frac{a_{\sup}}{b_{\inf}}|\Omega|\Big{)}.$

Note that

$u(t,x;s,u_{0}),v(t,x;s,u_{0})>0\quad\forall\,x\in\Omega,\,\,t\in(s,\infty).$

A positive entire solution of (1.1) is a solution $(u(t,x),v(t,x))$ which is
defined for all $t\in\mathbb{R}$ and
$\inf_{t\in\mathbb{R},x\in\Omega}u(t,x)>0$.

The first main result of the current paper is on the boundedness of
$\int_{\Omega}u^{-p}$ and $\int_{\Omega}u^{q}$ of the globally defined
classical solutions of (1.1).

######  Theorem 1.1 (Boundedness of $\int_{\Omega}u^{-p}$ and
$\int_{\Omega}u^{q}$).

Assume that $(1.5)^{\prime}$ holds. Then the following hold.

  * (1)

Let $p=1$, $\gamma=a_{\inf}-a_{\chi,\mu}$,
$M_{1}(p)=\frac{b_{\sup}|\Omega|}{a_{\inf}-a_{\chi,\mu}}$. Then for any
$u_{0}$ satisfying $(1.2)^{\prime}$, $s\in\mathbb{R}$ and $\tau>s$,

$\displaystyle\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq
e^{-\gamma(t-\tau)}\int_{\Omega}u^{-p}(\tau,x;s,u_{0})dx+M_{1}(p)+\tilde{M}_{1}(p,u_{0},\tau)\quad\forall\,t\geq\tau,$

and

$\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq\max\left\\{\int_{\Omega}u^{-p}(\tau,x;s,u_{0})dx,M_{1}(p)+\tilde{M}_{1}(p,\tau,s,u_{0})\right\\}\,\,\,\forall\,\,t\geq\tau,$
(1.10)

as well as

$\lim_{t-s\to\infty}\int_{\Omega}u^{-p}(t,x;s,u_{0})\leq M_{1}(p).$ (1.11)

  * (2)

Let $p>0$ be as in (1). There are $q>\max\\{2,N\\}$ and $M_{2}(p,q)>0$ such
that for any $s\in\mathbb{R}$, $u_{0}$ satisfying $(1.2)^{\prime}$, and
$\tau>s$, there is $\tilde{M}_{2}(p,q,\tau,s,u_{0})$ such that

$\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq
e^{-(t-\tau)}\int_{\Omega}u^{q}(\tau,x;s,u_{0})dx+M_{2}(p,q)+\tilde{M}_{2}(p,q,\tau,s,u_{0})\quad\forall\,t\geq\tau,$

and

$\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq\max\left\\{\int_{\Omega}u^{q}(\tau,x;s,u_{0})dx,M_{2}(p,q)+\tilde{M}_{2}(p,q,\tau,s,u_{0})\right\\}\quad\forall\,t\geq\tau,$

as well as

$\limsup_{t-s\to\infty}\int_{\Omega}u^{q}(t,x;s,u_{0})ds\leq M_{2}(p,q).$

######  Remark 1.1.

  * (1)

Theorem 1.1(1) improves boundedness result of
$\int_{\Omega}u^{-p}(t,x;s,u_{0})dx$ in [25, Lemma 3.4] (see (iii) in the
above) in the following two aspects. First, it provides some concrete
estimates for $\int_{\Omega}u^{-p}(t,x;s,u_{0})dx$. Second, it provides an
ultimate upper bound independent of $u_{0}$ for
$\int_{\Omega}u^{-p}(t,x;s,u_{0})dx$.

  * (2)

Theorem 1.1(2) improves the boundedness result of
$\int_{\Omega}u^{q}(t,x;s,u_{0})dx$ in [25, Theorem 1.1(2)] (see (ii) in the
above) in the following two aspects. First, it provides some concrete
estimates for $\int_{\Omega}u^{q}(t,x;s,u_{0})dx$. Second, it provides an
ultimate upper bound independent of $u_{0}$ for
$\int_{\Omega}u^{q}(t,x;s,u_{0})dx$.

  * (3)

The inequality (1.11) provides some improvement of (1.8) proved in [25]. To be
more precise, let $p$ be as in Theorem 1.1(1). By the Hölder’s inequality, for
any $s\in\mathbb{R}$ and $u_{0}$ satisfying $(1.2)^{\prime}$, we have

$\displaystyle|\Omega|$
$\displaystyle=\int_{\Omega}u^{\frac{p}{p+1}}(t,x;s,u_{0})u^{-\frac{p}{p+1}}(t,x;s,u_{0})dx$
$\displaystyle\leq\left(\int_{\Omega}u(t,x;s,u_{0})dx\right)^{\frac{p}{p+1}}\left(\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\right)^{\frac{1}{p+1}}\quad\forall\,t>s.$

Then by (1.11),

$\lim_{t-s\to\infty}\int_{\Omega}u(t,x;s,u_{0})dx\geq\frac{|\Omega|^{\frac{p+1}{p}}}{(2M_{1}(p))^{\frac{1}{p}}},$
(1.12)

which provides an ultimate lower bound independent of $u_{0}$ for
$\int_{\Omega}u(t,x;s,u_{0})dx$ and hence improves (1.8) proved in [25].

  * (4)

The upper bounds of $\int_{\Omega}u^{-p}(t,x;s,u_{0})dx$ and
$\int_{\Omega}u^{q}(t,x;s,u_{0})dx$ obtained in Theorem 1.1 provide some
useful tool for the study of the asymptotic behavior of globally defined
positive solutions of (1.1) (see Theorems 1.2-1.4 in the following).

The second main result of this paper is on the existence of globally absorbing
sets, which provides some insight on the asymptotic behavior of globally
defined positive solutions of (1.1).

######  Theorem 1.2 (Globally absorbing rectangle).

Assume that $(1.5)^{\prime}$ holds. Let $q>\max\\{2,N\\}$ and $p>0$ be as in
Theorem 1.1. There are $M_{0}^{*}>0$, $M_{1}^{*},M_{2}^{*}>0$ such that the
following hold.

  * (1)

The set

$\mathcal{E}=\left\\{u\in C^{0}(\bar{\Omega})\,|\,u\geq
0,\,\,{\int_{\Omega}u(x)dx\leq M_{0}^{*}},\int_{\Omega}u^{-p}(x)dx\leq
M_{1}^{*},\,\,\int_{\Omega}u^{q}(x)dx\leq M_{2}^{*}\right\\}$ (1.13)

is an invariant set of (1.1) in the sense that for any $u_{0}\in\mathcal{E}$
and $s\in\mathbb{R}$, $u_{0}$ satisfies $(1.2)^{\prime}$ and
$u(t,\cdot;s,u_{0})\in\mathcal{E}$ for all $t\geq s$. Moreover, for any
$0<\theta<1-\frac{2N}{q}$ and $\tau>0$, there is $M_{3}^{*}(\theta,\tau)>0$
such that for any $u_{0}\in\mathcal{E}$ and $s\in\mathbb{R}$,

$\|u(t,\cdot;s,u_{0})\|_{C^{\theta}(\bar{\Omega})}\leq
M_{3}^{*}(\theta,\tau)\quad\forall\,t\geq s+\tau.$ (1.14)

  * (2)

The set $\mathcal{E}$ is globally absorbing in the sense that for any $u_{0}$
satisfying $(1.2)^{\prime}$ and $s\in\mathbb{R}$,

$\begin{cases}\displaystyle\limsup_{t-s\to\infty}\int_{\Omega}u(t,x;s,u_{0})\leq
M_{0}^{*}\cr\displaystyle\limsup_{t-s\to\infty}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq
M_{1}^{*}\cr\displaystyle\limsup_{t-s\to\infty}\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq
M_{2}^{*}.\end{cases}$ (1.15)

Moreover, for any $0<\theta<1-\frac{2N}{q}$, there is $M_{4}^{*}(\theta)>0$
such that for any $u_{0}$ satisfying $(1.2)^{\prime}$, there is $T(u_{0})>0$
such that

$\|u(t,\cdot;s,u_{0})\|_{C^{\theta}(\bar{\Omega})}\leq
M_{4}^{*}(\theta)\quad\forall\,s\in\mathbb{R},\,\,t\geq s+T(u_{0}).$ (1.16)

######  Remark 1.2.

  * (1)

Assume that $(1.5)^{\prime}$ holds. (1.16) provides some improvement of (1.9)
proved in [25]. In fact, by (1.16), for any $s\in\mathbb{R}$ and $u_{0}$
satisfying $(1.2)^{\prime}$,

$\|u(t,\cdot;s,u_{0})\|_{\infty}\leq M_{4}^{*}(\theta)\quad\forall t-s\gg 1,$
(1.17)

which provides an ultimate upper bound independent of $u_{0}$ for
$\|u(t,\cdot;s,u_{0})\|_{\infty}$ and hence improves (1.9) proved in [25].

  * (2)

Theorem 1.2(2) implies that $\mathcal{E}$ is both pullback absorbing and
forward absorbing in the sense that for any $u_{0}$ satisfying
$(1.2)^{\prime}$,

$\begin{cases}\displaystyle\limsup_{s\to-\infty}\int_{\Omega}u(t,x;s,u_{0})dx\leq
M_{0}^{*}\quad&\forall\,\,t\in\mathbb{R}\cr\displaystyle\limsup_{s\to-\infty}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq
M_{1}^{*}\quad&\forall\,\,t\in\mathbb{R}\cr\displaystyle\limsup_{s\to-\infty}\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq
M_{2}^{*}\quad&\forall\,\,t\in\mathbb{R},\end{cases}$

and

$\begin{cases}\displaystyle\limsup_{t\to\infty}\int_{\Omega}u(t,x;s,u_{0})\leq
M_{0}^{*}\quad&\forall\,\,s\in\mathbb{R}\cr\displaystyle\limsup_{t\to\infty}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq
M_{1}^{*}\quad&\forall\,\,s\in\mathbb{R}\cr\displaystyle\limsup_{t\to\infty}\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq
M_{2}^{*}\quad&\forall\,\,s\in\mathbb{R},\end{cases}$

respectively.

  * (3)

Note that the existence of bounded absorbing sets and eventual initial-
independent boundedness of positive solutions are strongly related. It should
be pointed out that the existence of absorbing sets and eventual boundedness
of positive solutions have been studied in various logistic chemotaxis models.
For example, in [39], Winkler studied the existence of bounded absorbing sets
in $L^{\infty}$ for the following parabolic-elliptic chemotaxis model with
regular sensitivity and logistic type source,

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+g(u),\quad&x\in\Omega\cr 0=\Delta v-v+u,\quad&x\in\Omega\cr\frac{\partial
u}{\partial n}=\frac{\partial v}{\partial
n}=0,\quad&x\in\partial\Omega,\end{cases}$

where $g(u)=Au-Bu^{\alpha}$ for some $A,B>0$ and $\alpha>1$. We refer the
reader to [34], [35] and references therein for the study of eventual
boundedness of positive solutions of the following parabolic-parabolic
chemotaxis model,

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+g(u),\quad&x\in\Omega\cr v_{t}=\Delta
v-v+u,\quad&x\in\Omega\cr\frac{\partial u}{\partial n}=\frac{\partial
v}{\partial n}=0,\quad&x\in\partial\Omega,\end{cases}$

where $g(u)$ is a logistic type nonlinear function.

The following theorem is on the pointwise persistence, which is strongly
related to (1.12) and provides some further insight on the asymptotic behavior
of globally defined positive solutions of (1.1).

######  Theorem 1.3 (Uniform pointwise persistence).

Assume that $(1.5)^{\prime}$ holds. There is $m^{*}>0$ such that for any
$u_{0}$ satisfying $(1.2)^{\prime}$ and $s\in\mathbb{R}$,

$\liminf_{t-s\to\infty}\inf_{x\in\Omega}u(t,x;s,u_{0})\geq m^{*}.$ (1.18)

######  Remark 1.3.

The pointwise persistence (1.18) is proved for the first time for chemotaxis
models with singular sensitivity and logistic source, and implies the mass
persistence. We point out that mass persistence for chemotaxis systems with
regular chemotaxis sensitivity and logistic source was studied in [32] and
pointwise persistence was studied in [20, 21].

By (1.17) and (1.18), under the assumption $(1.5)^{\prime}$, any globally
defined positive solution with initial condition $u_{0}$ satisfying
$(1.2)^{\prime}$ is eventually bounded above and below by some positive
constants independent of its initial condition. It is natural to ask whether
globally defined positive solutions converge to some positive entire solution.
When $a(t,x)\equiv a$ and $b(t,x)\equiv b$, it is clear that
$(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ is a positive entire solution of
(1.1). When $a(t,x)$ and $b(t,x)$ depend on $t$ and $x$, it is not clear at
all whether (1.1) has positive entire solutions. The last main result of the
current paper is on the existence of positive entire solutions of (1.1).

######  Theorem 1.4 (Existence of positive entire solutions).

Suppose that $(1.5)^{\prime}$ holds. Let $\mathcal{E}$ be as in Theorem 1.2.
Then the followings hold.

  * (1)

There is a positive entire solution $(u^{*}(t,x),v^{*}(t,x))$ of (1.1) with
$u^{*}(t,\cdot)\in\mathcal{E}$ for any $t\in\mathbb{R}$.

  * (2)

If there is $T>0$ such that $a(t+T,x)=a(t,x)$ and $b(t+T,x)=b(t,x)$, then
(1.1) has a positive $T$-periodic solution $(u,v)=(u^{*}(t,x),v^{*}(t,x))$
with $u^{*}(t,\cdot)\in\mathcal{E}$ for any $t\in\mathbb{R}$.

  * (3)

If $a(t,x)\equiv a(x)$ and $b(t,x)\equiv b(x)$, then (1.1) has a positive
stationary solution $(u,v)=(u^{*}(x),v^{*}(x))$ with
$u^{*}(\cdot)\in\mathcal{E}$.

######  Remark 1.4.

As stated in (c), when $N=2$ and $a(t,x)\equiv a$, $b(t,x)\equiv b$, the
constant entire solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ is
exponentially stable under the assumption (1.4) and some other assumption (see
[6, Theorem 1]). By the arguments of [6, Theorem 1], it can also be proved
that, for general $N\geq 1$, $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ is
exponentially stable under the assumption (1.4) and some other assumption. But
when $a(t,x),b(t,x)$ are not constant functions, the arguments of [6, Theorem
1] are difficult to apply. We plan to study the stability of positive entire
solutions of (1.1) somewhere else.

It should be pointed out that a considerable amount of research has also been
carried out toward the finite-time blow-up or global boundedness of solutions
of the following parabolic-parabolic chemotaxis model with singular
sensitivity,

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla
v)+u(a-bu),\quad&x\in\Omega,\cr v_{t}=\Delta
v-v+u,\quad&x\in\Omega,\cr\frac{\partial u}{\partial n}=\frac{\partial
v}{\partial n}=0,\quad&x\in\partial\Omega.\end{cases}$ (1.19)

For example, for the case that $a=b=0$, it is proved that if
$0<\chi<\sqrt{\frac{2}{N}}$, then for any initial data $u_{0}\in
C^{0}(\bar{\Omega})$ and $v_{0}\in W^{1,\infty}(\Omega)$ with $u_{0}\geq 0$
and $v_{0}>0$ in $\bar{\Omega}$, there exists a global-in-time classical
solution of (1.19) (see [37]), and moreover, the global-in-time classical
solution is bounded (see [9]). When $\Omega$ is a smooth, bounded, convex two-
dimensional domain, it is shown in [26] that there is $\chi_{0}>1$ such that
for any initial data $u_{0}\in C^{0}(\bar{\Omega})$ and $v_{0}\in
W^{1,\infty}(\Omega)$ with $u_{0}\geq 0$ and $v_{0}>0$ in $\bar{\Omega}$,
(1.19) has a global bounded solution for $\chi\in(0,\chi_{0})$, which implies
that $0<\chi<1$ is not critical for the global existence of (1.19) on two-
dimensional domains. See [10, 27, 37] for more results.

For the case that $a,b>0$, it is proved that if $N=2$, then for any initial
data $0\leq u_{0}\in L^{2}(\Omega)$ and $0<v_{0}\in
H^{1+\epsilon_{0}}(\Omega)$ with $\inf_{x\in\Omega}v_{0}(x)>0$ and
$\epsilon_{0}\in(0,\frac{1}{2})$, (1.19) has a global classical solution (see
[1, 40]), and if (1.4) holds with $\mu=1$ and $a_{\inf}=a$, then globally
defined positive solutions of (1.19) are bounded (see [40]) and the constant
solution $(\frac{a}{b},\frac{a}{b})$ is exponentially stable (see [42, Theorem
1.1]). See [41] for the existence of weak solutions of (1.19) with $a,b>0$ in
the case $N\geq 3$. However, there is little study on the global existence of
classical solutions of (1.19) when $N\geq 3$.

We also point out that a considerable amount of research has been carried out
toward the finite-time blow-up or global boundedness of solutions of the
following chemotaxis model with regular chemotaxis sensitivity and logistic
source,

$\begin{cases}u_{t}=\Delta u-\chi\nabla\cdot(u\nabla
v)+u(a-bu),\quad&x\in\Omega,\cr\tau v_{t}=\Delta v-\mu v+\nu
u,\quad&x\in\Omega,\cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial
n}=0,\quad&x\in\partial\Omega.\end{cases}$ (1.20)

For example, when $\tau=0$ and $\mu=\nu=1$, it is proved in [33] that, if
$N\leq 2$ or $b>\frac{N-2}{N}\chi$, then for every nonnegative initial data
$u_{0}\in C^{0}(\bar{\Omega})$, (1.20) possesses a global bounded classical
solution which is unique. It should be pointed out that, when $a=b=0$ and
$N\geq 2$, finite-time blow-up of positive solutions occurs under some
condition on the mass and the moment of the initial data (see [17], [18],
[28], [29]). Hence the finite time blow-up phenomena in (1.20) is suppressed
to some extent by the logistic source. But it remains open whether in any
space dimensional setting, for every nonnegative initial data $u_{0}\in
C^{0}(\bar{\Omega})$ (1.20) possesses a unique global classical solution for
every $\chi>0$ and every $b>0$. It should be pointed out that finite-time
blow-up occurs in various variants of (1.20), for example, it occurs in (1.20)
with $\tau=0$, with the logistic source being replaced by logistic-type
superlinear degradation (see [31, 38]), and/or with the second equation being
replaced by the following one

$0=\Delta v-\frac{1}{|\Omega|}\int_{\Omega}u(\cdot,t)+u,\quad x\in\Omega$

(see [5, 7, 8, 36]).

The rest of this paper is organized in the following way. In section 2, we
present some lemmas to be used in later sections. In section 3, we study the
boundedness of $\int_{\Omega}u^{-p}$ and $\int_{\Omega}u^{q}$ of globally
defined positive solutions of (1.1) and prove the Theorem 1.1. In section 4,
we investigate the existence of absorbing invariant sets and the ultimate
upper and lower bounds of globally defined positive solutions and prove
Theorems 1.2 and 1.3. In section 5, we explore the existence of positive
entire solutions of (1.1) and prove the Theorem 1.4.

##  2 Preliminaries

In this section, we present some lemmas to be used in later sections.
Throughout this section, $\mu>0$ and $\nu>0$ are fixed. $C$ denotes some
generic constant which is independent of solutions, but may not be the same at
different places.

First, for convenience, we present two lemmas on fixed point theorems.

######  Lemma 2.1.

If $F$ is a closed subset of a Banach space $X,$ $G$ is a subset of a Banach
space $Y,$ $T_{y}:F\to F,$ $y$ in $G$ is a uniform contraction on $F$ and
$T_{y}x$ is continuous in $y\in G$ for each fixed $x$ in $F,$ then the unique
fixed point $g(y)$ of $T_{y},$ $y$ in $G,$ is continuous in $y$.

###### Proof.

See [15, Theorem 3.2 in Chapter 0]. ∎

######  Lemma 2.2.

Let $G$ be a closed convex set in a Banach space $X$ and let $T$ be a
continuous mapping of $G$ into itself such that the image $TG$ is precompact.
Then $T$ has a fixed point.

###### Proof.

It follows from [14, Corollary 11.2]. ∎

Next, we present two lemmas on the lower and upper bounds of the solutions of

$\begin{cases}\Delta v-\mu v+\nu u=0,\quad&x\in\Omega,\cr\frac{\partial
v}{\partial n}=0,\quad\quad\quad&x\in\partial\Omega.\end{cases}$ (2.1)

For given $u\in L^{p}(\Omega)$, let $v(\cdot;u)$ be the solution of (2.1).

######  Lemma 2.3.

Suppose that $u\in C^{0}(\bar{\Omega})$ is nonnegative and $\int_{\Omega}u>0$.
Then

$v(x;u)\geq{\delta_{0}}\int_{\Omega}u>0\quad{\rm in}\quad\Omega$

for some positive constants $\delta_{0}$ depending only on $\Omega$.

###### Proof.

It follows from the arguments of [13, Lemma 2.1]. ∎

######  Lemma 2.4.

For any $p>1$, there exists $C_{p}>0$ such that

$\max\Big{\\{}\|v(\cdot;u)\|_{L^{p}(\Omega)},\|\nabla
v(\cdot;u)\|_{L^{p}(\Omega)}\Big{\\}}\leq
C_{p}\|u(\cdot)\|_{L^{p}(\Omega)}\quad\forall u\in L^{p}(\Omega).$

###### Proof.

It follows from $L^{p}$-estimates for elliptic equations (see [2, Theorem
12.1]). ∎

Now, we present some properties of the semigroup generated by $-\Delta+\mu I$
complemented with Neumann boundary condition on $L^{p}(\Omega)$. For given
${1<p<\infty}$, let ${X_{p}}=L^{p}(\Omega)$ and ${A_{p}}=-\Delta+\mu I$ with

$D(A_{p})=\left\\{u\in W^{2,p}(\Omega)\,|\,\frac{\partial u}{\partial
n}=0\quad\text{on }\,\partial\Omega\right\\}.$

It is well known that $A_{p}$ is a sectorial operator in $X_{p}$ and thus
generates an analytic semigroup $\left(e^{-A_{p}t}\right)_{t\geq 0}$ in
$X_{p}$ (see, for example, [2, Theorem 13.4]). Moreover $0\in\rho(A_{p})$ and

$\|e^{-A_{p}t}u\|_{X_{p}}\leq e^{{-\mu t}}\|u\|_{X_{p}}\quad{\rm for}\quad
t\geq 0\,\,\,{\rm and}\,\,\,u\in X_{p}.$

Let $X_{p}^{\alpha}=D(A_{p}^{\alpha})$ equipped with the graph norm
$\|u\|_{\alpha,p}:=\|u\|_{X_{p}^{\alpha}}=\|A_{p}^{\alpha}u\|_{L^{p}}$.

######  Lemma 2.5.

  * (i)

Let $p\in(1,\infty).$ For each $\beta\geq 0$, there is $C_{p,\beta}>0$ such
that for some $\gamma>0,$

$\|A_{p}^{\beta}e^{-A_{p}t}\|_{L^{p}(\Omega)}\leq
C_{p,\beta}t^{-\beta}e^{-\gamma t}\;\;\text{for}\;\;t>0.$

  * (ii)

If $m\in\\{0,1\\}$ and $q\in[p,\infty]$ are such that
$m-\frac{N}{q}<2\beta-\frac{N}{p}$, then

$X_{p}^{\beta}\hookrightarrow W^{m,q}(\Omega).$

  * (iii)

If $2\beta-\frac{N}{p}>\theta\geq 0$, then

$X_{p}^{\beta}\hookrightarrow C^{\theta}(\Omega).$

###### Proof.

(i) It follows from [16, Theorem 1.4.3].

(ii) It follows from [16, Theorem 1.6.1].

(iii) It also follows from [16, Theorem 1.6.1]. ∎

######  Lemma 2.6.

Let $\beta\geq 0,$ $p\in(1,\infty)$. Then for any $\epsilon>0$ there exists
$C_{p,\beta,\epsilon}>0$ such that for any $w\in C^{\infty}_{0}(\Omega)$ we
have

$\|A_{p}^{\beta}e^{-tA_{p}}\nabla\cdot w\|_{L^{p}(\Omega)}\leq
C_{p,\beta,\epsilon}t^{-\beta-\frac{1}{2}-\epsilon}e^{-\gamma
t}\|w\|_{L^{p}(\Omega)}\quad\text{for all}\,\,t>0\,\text{and some
}\,\gamma>0.$ (2.2)

Accordingly, for all $t>0$ the operator $A_{p}^{\beta}e^{-tA_{p}}\nabla\cdot$
admits a unique extension to all of $L^{p}(\Omega)$ which is again denoted by
$A_{p}^{\beta}e^{-tA_{p}}\nabla\cdot$ and satisfies $\eqref{001}$ for all
$\mathbb{R}^{n}$-valued $w\in L^{p}(\Omega).$

###### Proof.

It follows from [19, Lemma 2.1]. ∎

Finally, we present some basic properties of solutions of (1.1). In the rest
of this paper, if no confusion occurs, we put $A=A_{p}$ for some $1<p<\infty$.

######  Lemma 2.7.

Assume $(1.5)^{\prime}$. For any given $u_{0}\in C^{0}(\bar{\Omega})$
satisfying ${(1.2)^{\prime}}$ and $s\in\mathbb{R}$, there is a unique
classical solution $(u(t,x;s,u_{0}),v(t,x;s,v_{0}))$ of (1.1) on $(s,\infty)$
with initial condition $u(s,x;s,u_{0})=u_{0}(x)$ (i.e.
$T_{\max}(s,u_{0})=\infty$). Moreover, $(u(t,x;s,u_{0}),v(t,x;s,u_{0}))$
satisfies

$\displaystyle u(t,\cdot;s,u_{0})$
$\displaystyle=e^{-A(t-s)}u_{0}-\chi\int_{s}^{t}e^{-A(t-\tau)}\nabla\cdot\left(\frac{u(\tau,\cdot;s,u_{0})}{v(\tau,\cdot;s,u_{0})}\nabla
v(\tau,\cdot;s,u_{0})\right)d\tau$
$\displaystyle\quad+\int_{s}^{t}e^{-A(t-\tau)}u(\tau,\cdot;s,u_{0})\big{[}{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u(\tau,\cdot;s,u_{0})\big{]}d\tau$
(2.3)

for any $t>s$, and if $\inf_{x\in\Omega}u_{0}(x)>0$, then for any $p>0$,

$\lim_{t\to
s+}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx=\int_{\Omega}u^{-p}_{0}(x)dx.$ (2.4)

###### Proof.

First, by the arguments of [12, Lemma 2.2] and [25, Theorem 1.2], for any
given $u_{0}\in C^{0}(\bar{\Omega})$ satisfying (1.2) and $s\in\mathbb{R}$,
there is a unique classical solution $(u(t,x;s,u_{0}),v(t,x;s,v_{0}))$ of
(1.1) on $(s,\infty)$ with initial condition $u(s,x;s,u_{0})=u_{0}(x)$ and
$(u(t,\cdot;s,u_{0}),v(t,\cdot;s,v_{0}))$ satisfies (2.7).

Next, if $\inf_{x\in\bar{\Omega}}u_{0}(x)>0$, by (1.3),

$\lim_{t\to s+}u^{-p}(t,x;s,u_{0})=u^{-p}_{0}(x)\quad\text{uniformly
in}\,\,x\in\bar{\Omega}.$

This implies (2.4) holds. The lemma is thus proved. ∎

To indicate the dependence of the classical solution
$(u(t,x;s,u_{0}),v(t,x;s,u_{0}))$ of (1.1) on $a(t,x)$ and $b(t,x)$, we may
put

$(u(t,x;s,u_{0},a,b),v(t,x;s,u_{0},a,b))=(u(t,x;s,u_{0}),v(t,x;s,u_{0})).$

######  Lemma 2.8.

Fix $s\in\mathbb{R}$. Let $u_{n},u_{0}\in C^{0}(\bar{\Omega})$ with
$u_{n}(x)\geq 0$ and $a_{n}(t,x),b_{n}(t,x)$ satisfy (H). If $u_{0}$ satisfies
${(1.2)^{\prime}}$, $\lim_{n\to\infty}\|u_{n}-u_{0}\|_{\infty}=0$, and

$\lim_{n\to\infty}\sup_{x\in\Omega}|a_{n}(t,x)-a_{0}(t,x)|=0,\quad\lim_{n\to\infty}\sup_{x\in\Omega}|b_{n}(t,x)-b_{0}(t,x)|=0$

locally uniformly in $t\in\mathbb{R}$ for some $a_{0}(t,x)$ and $b_{0}(t,x)$,
then $a_{0}(t,x),b_{0}(t,x)$ satisfy (H), there is $K>0$ such that $u_{n}$
satisfies (1.2) for $n\geq K$, and

$\lim_{n\to\infty}\|u(t,\cdot;s,u_{n},a_{n},b_{n})-u(t,\cdot;s,u_{0},a_{0},b_{0})\|_{\infty}=0\,\,{\rm
uniformly\,\,in}\,\,t\in[s,s+T]$ (2.5)

for any $T>0$.

###### Proof.

This lemma is about the continuity of solutions of (1.1) with respect to
initials and the coefficients in the equations. Note that solutions of (1.1)
satisfy the integral equation (2.7). It then suffices to prove the continuity
of solutions of (2.7) with respect to initials and the coefficients in the
equations.

First of all, it is clear that $a_{0}(t,x)$ and $b_{0}(t,x)$ satisfy (H).
Assume that $u_{0}\in C^{0}(\bar{\Omega})$ satisfies (1.2). For given $r>0$,
define

$\mathcal{B}(u_{0},r)=\\{u\in C^{0}(\bar{\Omega})\,|\,\|u-u_{0}\|_{\infty}\leq
r\\}.$

Fix $0<r\ll 1$ such that

$\int_{\Omega}\tilde{u}_{0}(x)dx\geq\frac{2}{3}\int_{\Omega}u_{0}(x)dx\quad\text{for
any}\,\,\tilde{u}_{0}\in\mathcal{B}(u_{0},r).$ (2.6)

Since $\lim_{n\to\infty}\|u_{n}-u_{0}\|_{\infty}=0$, there is $K>0$ such that
$u_{n}\in\mathcal{B}(u_{0},r)$ and
$\int_{\Omega}u_{n}(x)dx\geq\frac{2}{3}\int_{\Omega}u_{0}(x)dx>0$ for $n\geq
K$. This implies that $u_{n}$ satisfies (H) for $n\geq K$. Without loss of
generality, we may assume that $u_{n}\in\mathcal{B}(u_{0},r)$ for all $n\geq
1$. Then $\int_{\Omega}u_{n}(x)dx>0$ for all $n\geq 1$. By Lemma 2.7,
$u(t,x;s,u_{n},a_{n},b_{n})$ and $v(t,x;s,u_{0},a_{0},b_{0})$ are defined for
$t\geq s$.

Next, we note that it suffices to prove that (2.5) holds for $0<T\ll 1$. To
prove this, set

$\varepsilon:=\frac{\delta_{0}}{2}\int_{\Omega}u_{0}(x)dx>0,$

where $\delta_{0}$ is as in Lemma 2.3. For given $T>0$ and
$R>r+\|u_{0}\|_{C^{0}(\bar{\Omega})},$ we define the Banach space

$\mathcal{X}_{T}=C^{0}([s,s+T],C^{0}(\bar{\Omega}))$

equipped with the norm

$\|u\|_{\mathcal{X}_{T}}=\max_{s\leq t\leq
s+T}\|u(t,x)\|_{{C^{0}}(\bar{\Omega})}.$

Set

$\mathcal{S}(T)=\left\\{u\in\mathcal{X}_{T}:\|u\|_{\mathcal{X}_{T}}\leq
R,\;\text{and}\;{\nu A^{-1}u}\geq\varepsilon\;\text{for
all}\;t\in[s,s+T]\right\\},$

and

$\displaystyle\mathcal{Y}(T)=\\{a(\cdot,\cdot)\in\mathcal{X}_{T}\,:\,|a(t,x)-a(t^{{}^{\prime}},x)|\leq
B_{1}|t-t^{{}^{\prime}}|^{\gamma},\,\,\alpha\leq a(t,x)\leq
B_{2}\,\,\forall\,t,t^{{}^{\prime}}\in[s,s+T],\,\,x\in\bar{\Omega}\\},$

where $B_{1},B_{2},\alpha,\gamma$ are as in (H). It is clear that
$\mathcal{S}(T)$ and $\mathcal{Y}(T)$ are closed subsets of the Banach space
$\mathcal{X}_{T}$.

For $\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$ and
$(a,b)\in\mathcal{Y}(T)\times\mathcal{Y}(T)$, we define

$\displaystyle{(\mathcal{M}(\tilde{u}_{0},a,b)u)(t,\cdot)}$
$\displaystyle=e^{-A(t-s)}\tilde{u}_{0}-\chi\int_{s}^{t}e^{-A(t-\tau)}\nabla\cdot\left(\frac{u(\tau,\cdot)}{v(\tau,\cdot)}\nabla
v(\tau,\cdot)\right)d\tau$
$\displaystyle\quad+\int_{s}^{t}e^{-A(t-\tau)}u(\tau,\cdot)\big{[}{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u(\tau,\cdot)\big{]}d\tau,$

where $u\in\mathcal{S}(T)$ and $v(\tau,\cdot)={\nu}A^{-1}u(\tau,\cdot)$ for
$\tau\in[s,s+T]$. It is not difficult to prove that
$\mathcal{M}(\tilde{u}_{0},a,b)u\in\mathcal{X}_{T}$ for any
$u\in\mathcal{S}(T)$.

We claim that $\mathcal{M}(\tilde{u}_{0},a,b)$ maps $\mathcal{S}(T)$ into
itself for $0<T\ll 1$. To see this, let $p$, $\beta$, and $\epsilon>0$ be such
that $N<p$, $\frac{N}{2p}<\beta<\frac{1}{2}$, and
$\epsilon\in(0,\frac{1}{2}-\beta).$ Let $C_{p}$, $C_{p,\beta}$, and
$C_{p,\beta,\epsilon}$ be as in by Lemma 2.4, Lemma 2.5, and Lemma 2.6,
respectively. Then for any $\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$ and
$u\in\mathcal{S}(T)$, we have

$\displaystyle\|(\mathcal{M}(\tilde{u}_{0},a,b)u)(t,\cdot)\|_{C^{0}(\bar{\Omega})}$
$\displaystyle\leq\|e^{-A(t-s)}\tilde{u}_{0}\|_{C^{0}(\bar{\Omega})}+{\chi
C_{p,\beta}}\int_{s}^{t}\|A^{\beta}e^{-A(t-\tau)}\nabla\cdot\Big{(}\frac{u(\tau,\cdot)}{v(\tau,\cdot)}\nabla
v(\tau,\cdot)\Big{)}\|_{L^{p}(\Omega)}d\tau$
$\displaystyle\quad+{C_{p,\beta}}\int_{s}^{t}\|A^{\beta}e^{-A(t-\tau)}u(\tau,\cdot)\big{[}{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u(\tau,\cdot)\big{]}\|_{L^{p}(\Omega)}d\tau$
$\displaystyle\leq\|\tilde{u}_{0}\|_{C^{0}(\bar{\Omega})}+\frac{\chi}{\varepsilon}{C_{p,\beta}C_{p,\beta,\epsilon}C_{p}|\Omega|^{1/p}}R^{2}\int_{s}^{t}(t-\tau)^{-\beta-\frac{1}{2}-\epsilon}d\tau$
$\displaystyle\quad+{C^{2}_{p,\beta}|\Omega|^{1/p}}(R{\mu}+RB_{2}+R^{2}B_{2})\int_{s}^{t}(t-\tau)^{-\beta}d\tau$
$\displaystyle\leq\|u_{0}\|_{C^{0}(\bar{\Omega})}+r+\frac{\chi}{\varepsilon}{C_{p,\beta},C_{p,\beta,\epsilon}C_{p}|\Omega|^{1/p}}R^{2}T^{\frac{1}{2}-\beta-\epsilon}+{C^{2}_{p,\beta}|\Omega|^{1/p}}R({\mu}+B_{2}+RB_{2})T^{1-\beta}$

for all $t\in[s,s+T].$ We then have
$\|\mathcal{M}(u_{0},a,b)u\|_{\mathcal{X}_{T}}\leq R$ if $T\in(0,1)$ is
suitably small such that

$T\leq\left(\frac{R-r-\|u_{0}\|_{C^{0}(\bar{\Omega})}}{\frac{\chi}{\varepsilon}{C_{p,\beta}C_{p,\beta,\epsilon}C_{p}|\Omega|^{1/p}}R^{2}+{C^{2}_{p,\beta}|\Omega|^{1/p}}R({\mu}+B_{2}+RB_{2})}\right)^{\frac{1}{\frac{1}{2}-\beta-\epsilon}}.$
(2.7)

Note that

$\int_{\Omega}{(e^{-A(t-s)}}\tilde{u}_{0})(t,x)dx={e^{-(t-s)}\int_{\Omega}e^{(t-s)\Delta}\tilde{u}_{0}(x)dx\geq
e^{-T}}\int_{\Omega}\tilde{u}_{0}(x)dx,$
$\displaystyle\int_{\Omega}\left[\int_{s}^{t}{e^{-A(t-\tau)}}\nabla\cdot\Big{(}\frac{u}{v}\nabla
v\Big{)}d\tau\right]dx$
$\displaystyle={e^{-(t-s)}}\int_{s}^{t}\left[\int_{\Omega}e^{(t-\tau)\Delta}\nabla\cdot\Big{(}\frac{u}{v}\nabla
v\Big{)}dx\right]d\tau$
$\displaystyle={e^{-(t-s)}}\int_{s}^{t}\left[\int_{\Omega}\nabla\cdot\Big{(}\frac{u}{v}\nabla
v\Big{)}dx\right]d\tau$ $\displaystyle={e^{-(t-s)}}\int_{s}^{t}0d\tau=0,$

and

$\displaystyle\int_{\Omega}\left[\int_{s}^{t}{e^{-A(t-\tau)}}u\big{(}{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u\big{)}d\tau\right]dx$
$\displaystyle={e^{-(t-s)}}\int_{s}^{t}\left[\int_{\Omega}e^{(t-\tau)\Delta}u\big{(}{\mu}+a(\tau,\cdot)-b(\tau,\cdot)u\big{)}dx\right]d\tau$
$\displaystyle={e^{-(t-s)}}\int_{s}^{t}\left[\int_{\Omega}({\mu}+a(\tau,\cdot))u-b(\tau,\cdot)u^{2}dx\right]d\tau$
$\displaystyle\geq-B_{2}|\Omega|R^{2}T{e^{-T}}.$

It then follows that

$\displaystyle\int_{\Omega}(\mathcal{M}(\tilde{u}_{0},a,b)u)(t,x)dx$
$\displaystyle=\int_{\Omega}{e^{-A(t-s)}}\tilde{u}_{0}(x)dx+\int_{\Omega}\left[\int_{s}^{t}{e^{-A(t-\tau)}}\nabla\cdot\Big{(}\frac{u(\tau,x)}{v(\tau,x)}\nabla
v(\tau,x)\Big{)}d\tau\right]dx$
$\displaystyle\quad+\int_{\Omega}\left[\int_{s}^{t}{e^{-A(t-\tau)}}u(\tau,x)\big{(}{\mu}+a(\tau,x)-b(\tau,x)u(\tau,x)\big{)}d\tau\right]dx$
$\displaystyle\geq{e^{-T}}\int_{\Omega}\tilde{u}_{0}(x)dx-{e^{-T}}B_{2}|\Omega|R^{2}T\quad\forall
t\in[s,s+T].$

This together with (2.6) implies that, if

$0<{\frac{6T}{4-3e^{T}}}<\frac{1}{B_{2}|\Omega|R^{2}}\int_{\Omega}u_{0}(x)dx,$
(2.8)

then

$\int_{\Omega}(\mathcal{M}(\tilde{u}_{0},a,b)u)(t,x)dx\geq\frac{1}{2}\int_{\Omega}u_{0}(x)dx\quad\forall\,\tilde{u}_{0}\in\mathcal{B}(u_{0},r),\,\,(a,b)\in\mathcal{Y}_{0}(T),\,\,u\in\mathcal{S}(T).$

Hence

$({\nu}A^{-1}\mathcal{M}(\tilde{u}_{0},a,b)u)(t,x)\geq\delta_{0}\int_{\Omega}\mathcal{M}(\tilde{u}_{0},a,b)u(x,t)dx\geq\frac{\delta_{0}}{2}\int_{\Omega}u_{0}(x)dx=\varepsilon$

for all $t\in[s,s+T].$ Therefore, the claim holds with any $T>0$ satisfies
(2.7) and (2.8).

We then prove that the mapping $\mathcal{M}(\tilde{u}_{0},a,b)$ is a uniform
contraction on $\mathcal{S}(T)$ for $0<T\ll 1$. Let $p$, $\beta$, $\epsilon$,
and $C_{p},C_{p,\beta},C_{p,\beta,\epsilon}$ be as in the above. By similar
arguments as in the above, for given $u,w\in\mathcal{S}(T)$ with $v:=\nu
A^{-1}u,$ $\bar{v}:=\nu A^{-1}w,$ we have

$\displaystyle\|(\mathcal{M}(\tilde{u}_{0},a,b)u)(t,\cdot)-(\mathcal{M}(\tilde{u}_{0},a,b)w)(t,\cdot)\|_{C^{0}(\bar{\Omega})}$
$\displaystyle\leq\chi{C_{p,\beta}}\int_{s}^{t}\|{A^{\beta}e^{-A(t-\tau)}}\nabla\cdot\Big{(}\frac{u(\tau,\cdot)}{v(\tau,\cdot)}\nabla
v(\tau,\cdot)-\frac{w(\tau,\cdot)}{\bar{v}(\tau,\cdot)}\nabla\bar{v}(\tau,\cdot)\Big{)}\|_{L^{p}(\Omega)}d\tau$
$\displaystyle\quad+{C_{p,\beta}}\int_{s}^{t}\|{A^{\beta}e^{-A(t-\tau)}}(u(\tau,\cdot)-w(\tau,\cdot))\big{[}{\mu}+a(\tau,\cdot)-b(\tau,\cdot)(u(\tau,\cdot)+w(\tau,\cdot))\big{]}\|_{L^{p}(\Omega)}d\tau$
$\displaystyle\leq\frac{\chi{C_{p,\beta}C_{p,\beta,\epsilon}C_{p}|\Omega|^{1/p}}}{\varepsilon}\int_{s}^{t}(t-\tau)^{-\beta-\frac{1}{2}-\epsilon}\|u(\tau)-w(\tau)\|_{C^{0}(\Omega)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\cdot\Big{(}\|u(\tau)\|_{C^{0}(\Omega)}+\|w(\tau)\|_{C^{0}(\Omega)}+\frac{{C_{2p}}}{\varepsilon}\|w(\tau)\|^{2}_{C^{0}(\Omega)}\Big{)}d\tau$
$\displaystyle\quad+{C^{2}_{p,\beta}|\Omega|^{1/p}}\int_{s}^{t}(t-\tau)^{-\beta}\|u(\tau)-w(\tau)\|_{C^{0}(\Omega)}\Big{(}{\mu}+B_{2}+B_{2}\big{(}\|u(\tau)\|_{C^{0}(\Omega)}+\|w(\tau)\|_{C^{0}(\Omega)}\big{)}\Big{)}d\tau$
$\displaystyle\leq
T^{\frac{1}{2}-\beta-\epsilon}\Big{(}\frac{\chi{C_{p,\beta}C_{p,\beta,\epsilon}C_{p}|\Omega|^{1/2}}}{\varepsilon^{2}}(2\varepsilon
R+{C_{2p}}R^{2})+{C^{2}_{p,\beta}|\Omega|^{1/p}}({\mu}+B_{2}+2RB_{2})\Big{)}\|u-w\|_{\mathcal{X}}$
(2.9)

for all $t\in[s,s+T]$. It then follows that $\mathcal{M}(\tilde{u}_{0},a,b)$
is a uniform contraction on $\mathcal{S}(T)$ for any $T>0$ satisfying (2.7),
(2.8), and

$T<\left[\frac{\chi{C_{p,\beta}C_{p,\beta,\epsilon}C_{p}}|\Omega|^{1/p}}{\varepsilon^{2}}(2\varepsilon
R+{C_{2p}}R^{2})+{C_{p,\beta}^{2}|\Omega|^{1/p}}({\mu}+B_{2}+2RB_{2})\right]^{\frac{-1}{\frac{1}{2}-\beta-\epsilon}}.$
(2.10)

We now prove $\mathcal{M}({\tilde{u}_{0}},a,b)$ is uniformly continuous in
$\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$ and
$(a,b)\in\mathcal{Y}(T)\times\mathcal{Y}(T)$. Note that for any
$\tilde{u}_{1},\tilde{u}_{2}\in\mathcal{B}(u_{0},r)$,
$(a_{1},b_{1}),(a_{2},b_{2})\in\mathcal{Y}(T)\times\mathcal{Y}(T)$, and
$u\in\mathcal{S}(T)$,

$\displaystyle\mathcal{M}(\tilde{u}_{2},a_{2},b_{2})u-\mathcal{M}(\tilde{u}_{1},a_{1},b_{1})u=$
$\displaystyle[\mathcal{M}(\tilde{u}_{2},a_{2},b_{2})u-\mathcal{M}(\tilde{u}_{1},a_{2},b_{2})u]$
$\displaystyle+[\mathcal{M}(\tilde{u}_{1},a_{2},b_{2})u-\mathcal{M}(\tilde{u}_{1},a_{1},b_{2})u]$
$\displaystyle+[\mathcal{M}(\tilde{u}_{1},a_{1},b_{2})u-\mathcal{M}(\tilde{u}_{1},a_{1},b_{1})u].$

It then suffices to prove that $\mathcal{M}(\tilde{u}_{0},a,b)$ is uniformly
continuous in $\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$ (resp. uniformly
continuous in $a\in\mathcal{Y}(T)$, uniformly continuous in
$b\in\mathcal{Y}(T)$). Observe that, for any
$\tilde{u}_{n},\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$,
$(a,b)\in\mathcal{Y}_{0}(T)\times\mathcal{Y}_{0}(T)$, and
$u\in\mathcal{S}(T)$, by comparison principles for parabolic equations, we
have

$\|(\mathcal{M}(\tilde{u}_{n},a,b)u)(t,\cdot)-(\mathcal{M}({\tilde{u}_{0}},a,b)u)(t,\cdot)\|_{C^{0}(\bar{\Omega})}=\|e^{-(t-s)A}(\tilde{u}_{n}-\tilde{u}_{0})\|_{C^{0}(\bar{\Omega})}\leq\|\tilde{u}_{n}-\tilde{u}_{0}\|_{C^{0}(\bar{\Omega})}.$

Thus, the mapping $\mathcal{M}({\tilde{u}_{0}},a,b)$ is uniformly continuous
in $\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$. For any
$\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$, $a_{n},a_{0},b\in\mathcal{Y}(T)$, and
$u\in\mathcal{S}(T)$,

$\displaystyle\|(\mathcal{M}(\tilde{u}_{0},a_{n},b)u)(t,\cdot)-(\mathcal{M}(\tilde{u}_{0},a_{0},b)u)(t,\cdot)\|_{C^{0}(\bar{\Omega})}$
$\displaystyle=\|\int_{s}^{t}e^{-A(t-\tau)}u(\tau,\cdot)\big{[}a_{n}(\tau,\cdot)-a_{0}(\tau,\cdot)\big{]}d\tau\|_{C^{0}(\bar{\Omega})}$
$\displaystyle\leq R\|a_{n}-a_{0}\|_{\mathcal{X}_{T}}T.$

This implies that the mapping $\mathcal{M}({\tilde{u}_{0}},a,b)$ is uniformly
continuous in $a\in\mathcal{Y}(T)$. For any
$\tilde{u}_{0}\in\mathcal{B}(u_{0},r)$, $a,b_{n},b_{0}\in\mathcal{Y}(T)$, and
$u\in\mathcal{S}(T)$,

$\displaystyle\|(\mathcal{M}(\tilde{u}_{0},a,b_{n})u)(t,\cdot)-(\mathcal{M}(\tilde{u}_{0},a,b_{0})u)(t,\cdot)\|_{C^{0}(\bar{\Omega})}$
$\displaystyle=\|\int_{s}^{t}e^{-A(t-\tau)}u^{2}(\tau,\cdot)\big{[}b_{n}(\tau,\cdot)-b_{0}(\tau,\cdot)\big{]}d\tau\|_{C^{0}(\bar{\Omega})}$
$\displaystyle\leq R^{2}\|b_{n}-b_{0}\|_{\mathcal{X}_{T}}T.$

Therefore, the mapping $\mathcal{M}({\tilde{u}_{0}},a,b)$ is also uniformly
continuous in $b\in\mathcal{Y}(T)$.

Finally, Let $T>0$ satisfy (2.7), (2.8), and (2.10). By Lemmas 2.1 and 2.7, we
conclude that $\mathcal{M}({\tilde{u}_{0}},a,b)$ has a unique fixed point
$u\in\mathcal{S}(T)$ fulfilling
$(\mathcal{M}({\tilde{u}_{0}},a,b)u)(t,\cdot)=u(t,\cdot;s,{\tilde{u}_{0}},a,b)$
for $t\in[s,s+T],$ and we also have that $u(t,\cdot;s,{\tilde{u}_{0}},a,b)\in
C^{0}(\bar{\Omega})$ is continuous in ${\tilde{u}_{0}}\in\mathcal{B}(u_{0},r)$
and $(a,b)\in\mathcal{Y}(T)$ uniformly in $t\in[s,s+T].$ The lemma is thus
proved. ∎

##  3 Boundedness of positive solutions

In this section, we investigate the boundedness of positive solutions of
(1.1), and prove Theorem 1.1. Throughout this section, we assume that
$(1.5)^{\prime}$ holds.

For given $s\in\mathbb{R}$ and $u_{0}$ satisfying (1.2), put
$(u(t,x),v(t,x))=(u(t,x;s,u_{0}),v(t,x;s,u_{0}))$ for $t\geq s$ and
$x\in\bar{\Omega}$. Observe that, for any $1<q<\infty$,

$\displaystyle\frac{1}{q}\int_{\Omega}u_{t}^{q}(t,x)dx$
$\displaystyle=-(q-1)\int_{\Omega}u^{q-2}(t,x)|\nabla
u(t,x)|^{2}dx-\chi(q-1)\int_{\Omega}\frac{u^{q-1}(t,x)}{v(t,x)}\nabla
u(t,x)\cdot\nabla v(t,x)dx$
$\displaystyle\,\,\,\,+\int_{\Omega}a(t,x)u^{q}(t,x)-\int_{\Omega}b(t,x)u^{q+1}(t,x)dx\quad\forall\,{t>s},$

and for any $p>0$,

$\displaystyle\frac{1}{p}\cdot\frac{d}{dt}\int_{\Omega}u^{-p}(t,x)=$
$\displaystyle-(p+1)\int_{\Omega}u^{-p-2}(t,x)|\nabla
u(t,x)|^{2}+(p+1)\chi\int_{\Omega}\frac{u^{-p-1}(t,x)}{v(t,x)}\nabla
u(t,x)\cdot\nabla v(t,x)$
$\displaystyle-\int_{\Omega}a(t,x)u^{-p}(t,x)+\int_{\Omega}b(t,x)u^{-p+1}(t,x)\quad\forall\,{t>s}.$
(3.1)

In the rest of this section, we may omit $(t,x)$ inside the integrals if no
confusion occurs.

We first present some lemmas.

######  Lemma 3.1.

For any $s\in\mathbb{R}$ and $u_{0}$ satisfying ${(1.2)^{\prime}}$,

$\int_{\Omega}u(t,x)dx\leq m^{*}(\tau,s,u_{0})={\rm
max}\Big{\\{}\int_{\Omega}u(\tau,x)dx,\frac{a_{\sup}}{b_{\inf}}|\Omega|\Big{\\}}\quad\forall\,\,t>\tau\geq
s$

and

$\limsup_{t-s\to\infty}\int u(t,x)dx\leq\frac{a_{\sup}}{b_{\inf}}|\Omega|,$

where $|\Omega|$ is the Lebesgue measure of $\Omega$.

###### Proof.

By integrating the first equation in (1.1) with respect to $x$, we get that

$\displaystyle\frac{d}{dt}\int_{\Omega}u$ $\displaystyle=\int_{\Omega}\Delta
u-\chi\int_{\Omega}\nabla\cdot\Big{(}\frac{u}{v}\nabla
v\Big{)}+\int_{\Omega}a(x,t)u-b(x,t)u^{2}$
$\displaystyle=\int_{\Omega}a(x,t)u(x,t)dx-\int_{\Omega}b(x,t)u^{2}(x,t)dx$
$\displaystyle\leq
a_{\sup}\int_{\Omega}u-\frac{b_{\inf}}{|\Omega|}\Big{(}\int_{\Omega}u\Big{)}^{2}.$

This together with comparison principle for scalar ODEs implies that

$\int_{\Omega}u(t,x)dx\leq{\rm
max}\Big{\\{}\int_{\Omega}u(\tau,x)dx,\frac{a_{\sup}}{b_{\inf}}|\Omega|\Big{\\}}\quad\forall\,\,t>\tau\geq
s$

and

$\limsup_{t-s\to\infty}\int u(t,x)dx\leq\frac{a_{\sup}}{b_{\inf}}|\Omega|.$

The lemma is thus proved. ∎

######  Lemma 3.2.

Let $q\geq 3$ and $q-1\leq k<2q-2.$ There exist positive constants $M(q,k)>0$
and $M^{*}(q,k)>0$ such that

$\int_{\Omega}\frac{|\nabla v|^{2q}}{v^{k+1}}\leq
M(k,q)\int_{\Omega}\frac{u^{q}}{v^{k-q+1}}+M^{*}(k,q)\int_{\Omega}v^{2q-k-1}$

for any $s\in\mathbb{R}$, $u_{0}\in C^{0}(\bar{\Omega})$ satisfying
${(1.2)^{\prime}}$, and ${t\in(s,\infty)}$.

###### Proof.

It follows from [25, Proposition 1.3]. ∎

######  Lemma 3.3.

For any $p>0$, $s\in\mathbb{R}$ and $u_{0}\in C^{0}(\bar{\Omega})$ satisfying
${(1.2)^{\prime}}$,

$\int_{\Omega}u\geq|\Omega|^{\frac{p+1}{p}}\left(\int_{\Omega}u^{-p}\right)^{-\frac{1}{p}}\quad\forall\,t>s.$

###### Proof.

For any given $p>0$, by Hölder’s inequality, we have

$\displaystyle|\Omega|=\int_{\Omega}u^{\frac{p}{p+1}}u^{-\frac{p}{p+1}}\leq\Big{(}\int_{\Omega}u\Big{)}^{\frac{p}{p+1}}\Big{(}\int_{\Omega}u^{-p}\Big{)}^{\frac{1}{p+1}}\quad\forall\,t>s.$

This implies that

$\int_{\Omega}u\geq|\Omega|^{\frac{p+1}{p}}\left(\int_{\Omega}u^{-p}\right)^{-\frac{1}{p}}\quad\forall\,t>s.$

The lemma is thus proved. ∎

######  Lemma 3.4.

Let $p>0$. Then for every $\beta>0$, we have

$\displaystyle(p+1)\chi\int_{\Omega}\frac{u^{-p-1}}{v}\nabla u\cdot\nabla v$
$\displaystyle\leq(p+1)\int_{\Omega}u^{-p-2}|\nabla
u|^{2}+\frac{(p+1)\beta\mu}{p}\int_{\Omega}u^{-p}$
$\displaystyle\quad+\left[\frac{(p+1)(\chi-\beta)^{2}}{4}-\frac{(p+1)\beta}{p}\right]\int_{\Omega}u^{-p}\frac{|\nabla
v|^{2}}{v^{2}}$

for all $t\in(s,\infty)$.

###### Proof.

This is Lemma [25, Lemma 3.3]. ∎

######  Lemma 3.5.

Let $R>0$ be such that

$R>\begin{cases}\frac{\mu\chi^{2}}{4},&\text{if\,\, $0<\chi\leq 2$}\\\
\mu(\chi-1),&\text{if \,$\chi>2$.}\\\ \end{cases}$

Then there is $\beta>0$, $\beta\not=\chi$ such that

$\frac{(p+1)\beta\mu}{p}-R<0,$

where $p$ is given by

$p=\frac{4\beta}{(\chi-\beta)^{2}}.$

###### Proof.

This is [25, Lemma 3.1]. ∎

We now prove Theorem 1.1.

###### Proof of Theorem 1.1.

(1) It follows from the arguments of [25, Theorem 1.1(3)]. To be more precise,
let $\beta=\chi+2-2\sqrt{\chi+1}$. Then

$p:=\frac{4\beta}{(\chi-\beta)^{2}}=\frac{4(\chi+2-2\sqrt{\chi+1})}{(-2+2\sqrt{\chi+1})^{2}}=1\,\,\Longrightarrow\,\,\frac{(\chi-\beta)^{2}}{4}-\beta=0,$

and

$\frac{(p+1)\beta\mu}{p}=2\beta\mu=2(\chi+2-2\sqrt{\chi+1})\mu=a_{\chi,\mu}<a_{\inf}.$

By (3),

$\displaystyle\cdot\frac{d}{dt}\int_{\Omega}u^{-1}\leq-2\int_{\Omega}u^{-3}|\nabla
u|^{2}+2\chi\int_{\Omega}\frac{u^{-2}}{v}\nabla u\cdot\nabla
v-a_{\inf}\int_{\Omega}u^{-1}+b_{\sup}|\Omega|$

for all $t>s$. By Lemma 3.4, we have that

$\displaystyle\cdot\frac{d}{dt}\int_{\Omega}u^{-1}$ $\displaystyle\leq
2\left[\frac{(\chi-\beta)^{2}}{4}-{\beta}\right]\int_{\Omega}u^{-1}\frac{|\nabla
v|^{2}}{v^{2}}+\left[2\beta\mu-
a_{\inf}\right]\int_{\Omega}u^{-1}+b_{\sup}|\Omega|$
$\displaystyle=-(a_{\inf}-a_{\chi,\mu})\int_{\Omega}u^{-1}+b_{\sup}|\Omega|,\,\,\,\,\forall\,t>\tau>s.$

It then follows from comparison principle for scalar ODEs that

$\int_{\Omega}u^{-1}(t,x;s,u_{0})dx\leq
e^{-(a_{\inf}-a_{\chi,\mu})(t-\tau)}\int_{\Omega}u^{-1}(\tau,x;s,u_{0})dx+\frac{b_{\sup}|\Omega|}{a_{\inf}-a_{\chi,\mu}}$

and

$\int_{\Omega}u^{-1}(t,x;s,u_{0})dx\leq\max\Big{\\{}\int_{\Omega}u^{-1}(\tau,x;s,u_{0})dx,\frac{b_{\sup}|\Omega|}{a_{\inf}-a_{\chi,\mu}}\Big{\\}}$
(3.2)

for any $t>\tau>s$. Moreover,

$\lim_{t-s\to\infty}\int_{\Omega}u^{-1}(t,x;s,u_{0})dx=\frac{b_{\sup}|\Omega|}{a_{\inf}-a_{\chi,\mu}}.$

Theorem 1.1(1) thus follows.

(2) It also follows from the arguments of [25, Theorem 1.1(3)]. To be more
precise, by Lemmas 2.3 and 3.3, we have

$v(t,x)\geq\delta_{0}\int_{\Omega}u\geq\delta_{0}|\Omega|^{2}\left(\int_{\Omega}u^{-1}\right)^{-1}\quad\forall\,t>s.$

By (3.2) and the assumption $(1.2)^{\prime}$

$\int_{\Omega}u\geq\frac{(a_{\inf}-a_{\chi,\mu})\min\\{1,\chi\\}}{b_{\sup}|\Omega|},\quad
v\geq\frac{\delta_{0}(a_{\inf}-a_{\chi,\mu})\min\\{1,\chi\\}|\Omega|}{b_{\sup}}\quad\text{for
all}\,\,t\in[s+\tau_{0},\infty).$

By the arguments of [25, Theorem 1.1 (3)], for any $q>\max\\{2,N\\}$ and any
$\varepsilon>0$, there is $C(\varepsilon,q)>0$ such that

$\displaystyle\frac{1}{q}\cdot\frac{d}{dt}\int_{\Omega}u^{q}$
$\displaystyle\leq\frac{b_{\sup}|\Omega|(q-1)\Big{\\{}(C^{*})^{\frac{1}{q+1}}+\varepsilon\Big{\\}}\max\\{\chi,\chi^{2}\\}}{4\delta_{0}(a_{\inf}-a_{\chi,\mu})}\int_{\Omega}u^{q+1}$
$\displaystyle\quad+\frac{b_{\sup}|\Omega|(q-1)C(\varepsilon,p)\max\\{\chi,\chi^{2}\\}}{4\delta_{0}(a_{\inf}-a_{\chi,\mu})}\Big{(}\int_{\Omega}u\Big{)}^{{q+1}}$
$\displaystyle\quad+a_{\sup}\int_{\Omega}u^{q}-b_{\inf}\int_{\Omega}u^{q+1}\,\,\forall\,t\in(s+\tau_{0},\infty).$

By the assumption $(1.5)^{\prime}$, there are $q>\max\\{2,n\\}$ and
$0<\varepsilon=\varepsilon(q)\ll 1$ such that

$\frac{b_{\sup}|\Omega|(q-1)\Big{\\{}(C^{*}(n,\nu))^{\frac{1}{q+1}}+\varepsilon\Big{\\}}\max\\{\chi,\chi^{2}\\}}{4\delta_{0}(a_{\inf}-a_{\chi,\mu})}<b_{\inf}.$

Note that

$\int_{\Omega}u^{q+1}\geq\frac{1}{|\Omega|}\Big{(}\int_{\Omega}u^{q}\Big{)}^{\frac{q+1}{p}}\quad\text{for
all}\,\,t\in(s,\infty).$

Therefore, there is $b_{1}>0$ such that

$\displaystyle\frac{1}{q}\cdot\frac{d}{dt}\int_{\Omega}u^{q}$
$\displaystyle\leq\frac{b_{\sup}|\Omega|(q-1)C(\varepsilon,q)\max\\{\chi,\chi^{2}\\}}{4\delta_{0}(a_{\inf}-a_{\chi,\mu})}\Big{(}\int_{\Omega}u\Big{)}^{{q+1}}$
$\displaystyle\quad+a_{\sup}\int_{\Omega}u^{q}-b_{1}\Big{(}\int_{\Omega}u^{q}\Big{)}^{\frac{q+1}{p}}\quad\text{for
all}\,\,t\in(s+\tau_{0},\infty).$

This together with Lemma 3.1 implies that there are $M_{2}(1,q)$ and
$\tilde{M}_{2}(1,q,\tau,s,u_{0})$ such that

$\begin{cases}\frac{d}{dt}\int_{\Omega}u^{q}\leq-\int_{\Omega}u^{q}+M_{2}(1,q)\quad\forall\,t\geq
s+\tau_{0}\cr\frac{d}{dt}\int_{\Omega}u^{q}\leq-\int_{\Omega}u^{q}+M_{2}(1,q)+\tilde{M}_{2}(1,q,\tau,s,u_{0})\quad\forall\,t\geq\tau>s.\end{cases}$
(3.3)

Theorem 1.1(2) then follows. ∎

##  4 Globally attracting invariant rectangle and pointwise persistence

In this section, we investigate the existence of globally attracting invariant
rectangle for (1.1) and pointwise persistence, and prove Theorems 1.2 and 1.3.

We first prove a lemma.

######  Lemma 4.1.

Assume that ${(1.5)^{\prime}}$ holds. Let $p>0$ be as in Theorem 1.1(1). There
are $M_{1}^{*}>0$ and $\tilde{M}_{1}^{*}>0$ such that for every $u_{0}\in
C^{0}(\bar{\Omega})$ satisfying ${(1.2)^{\prime}}$ and
$\int_{\Omega}u_{0}(x)dx\leq\frac{a_{\sup}}{b_{\inf}}|\Omega|$,
$s\in\mathbb{R}$, and $\tau>0$,

$\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq\max\Big{\\{}\int_{\Omega}u^{-p}(s+\tau,x;s,u_{0})dx,M_{1}^{*}\Big{\\}}\quad\text{for
all}\,\,s+\tau<t<\infty$ (4.1)

and

$\limsup_{t-s\to\infty}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq M_{1}^{*},$
(4.2)
$\liminf_{t-s\to\infty}\int_{\Omega}u(t,x;s,u_{0})dx\geq\tilde{M}_{1}^{*},\quad\liminf_{t-s\to\infty}\inf_{x\in\Omega}v(t,x;s,u_{0})\geq\delta_{0}\tilde{M}_{1}^{*}.$
(4.3)

In addition, if $\inf_{x\in\Omega}u_{0}(x)>0$ and
$\int_{\Omega}u_{0}^{-p}(x)dx\leq M_{1}^{*}$, then

$\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq M_{1}^{*}\quad\forall\,\,s<t<\infty$
(4.4)

and

$\int_{\Omega}u(t,x;s,u_{0})\geq\tilde{M}_{1}^{*},\quad\inf_{x\in\Omega}v(t,x;s,u_{0})\geq\delta_{0}\tilde{M}_{1}^{*}\quad\forall
t\in[s,\infty).$ (4.5)

###### Proof.

First, let $M_{1}^{*}=M_{1}(p)$. Then, by Lemma 3.1,

$\int_{\Omega}u(\tau,x;s,u_{0})dx\leq\frac{a_{\sup}}{b_{\inf}}|\Omega|\quad\forall\,\,\tau\geq
s.$

Hence

$m^{*}(\tau,s,u_{0})=\frac{a_{\sup}}{b_{\inf}}\quad{\rm
and}\quad\tilde{M}_{1}(p,\tau,s,u_{0})=0\quad\forall\,\,\tau\geq s.$ (4.6)

This together with (1.10) and (1.11) implies (4.1) and (4.2) hold.

Next, by Hölder inequality, we have

$\displaystyle|\Omega|$
$\displaystyle=\int_{\Omega}u^{\frac{p}{p+1}}(t,x;s,u_{0})u^{-\frac{p}{p+1}}(t,x;s,u_{0})dx$
$\displaystyle\leq\big{(}\int_{\Omega}u(t,x;s,u_{0})dx\big{)}^{\frac{p}{p+1}}\big{(}\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\big{)}^{\frac{1}{p+1}}\quad\forall\,t>s.$
(4.7)

This together with (4.2) and Lemma 2.3 implies that (4.3) holds with
$\tilde{M}_{1}^{*}=\frac{|\Omega|^{\frac{p+1}{p}}}{(M_{1}^{*})^{\frac{1}{p}}}$.

Now, assume $\inf_{x\in\Omega}u_{0}(x)>0$ and
$\int_{\Omega}u_{0}^{-p}(x)dx\leq M_{1}^{*}$. By (2.4),

$\lim_{\tau\to
0}\int_{\Omega}u^{-p}(s+\tau,x;s,u_{0})dx=\int_{\Omega}u_{0}^{-p}(x)dx\leq
M_{1}^{*}.$

This together with (4.1) implies (4.4). (4.5) then follows from (4.4), (4),
and Lemma 2.3. The lemma is thus proved. ∎

Next, we prove Theorem 1.2.

###### Proof of Theorem 1.2.

Let $p$ and $M_{1}(p)$ be as in Theorem 1.1(1), and $q$ and $M_{2}(p,q)$ be as
in Theorem 1.1(2). Let $M_{0}^{*}=\frac{a_{\sup}}{b_{\inf}}|\Omega|$,
$M_{1}^{*}=M_{1}(p)$, and $M_{2}^{*}=M_{2}(p,q)$. In the following, we prove
that (1) and (2) in Theorem 1.2 hold with these $p,q,M_{1}^{*},M_{2}^{*}$ and
some $M_{3}^{*}(\tau,\theta)$, $M_{4}^{*}(\theta)$.

(1) We first prove that the set $\mathcal{E}$ is invariant.

First of all, for any $u_{0}\in\mathcal{E}$,

$|\Omega|^{2}=\left(\int_{\Omega}u_{0}^{-p/2}(x)u_{0}^{p/2}(x)dx\right)^{2}\leq\int_{\Omega}u_{0}^{-p}(x)dx\cdot\int_{\Omega}u_{0}^{p}(x)dx\leq
M_{1}^{*}\int_{\Omega}u_{0}^{p}(x)dx.$

This implies that $\int_{\Omega}u_{0}^{p}(x)dx>0$ and then
$\int_{\Omega}u_{0}(x)dx>0$. Recall that $p=1$ and

$M_{1}^{*}=M_{1}(p)=\frac{b_{\sup}|\Omega|}{a_{\inf}-a_{\chi,\mu}}\leq\frac{b_{\sup}|\Omega|\max\\{1,\frac{1}{\chi}\\}}{a_{\inf}-a_{\chi,\mu}},$

Hence $u_{0}$ satisfies $(1.2)^{\prime}$ with $\tau_{0}=0$.

For any $s\in\mathbb{R}$ and $u_{0}\in\mathcal{E}$, by Lemma 3.1,

$\int_{\Omega}u(t,x;s,u_{0})dx\leq\max\left\\{\int_{\Omega}u_{0}(x)dx,M_{0}^{*}\right\\}=M_{0}^{*}\quad\forall\,t\geq
s.$ (4.8)

By Lemma 4.1, there holds

$\int_{\Omega}u{{}^{-p}}(t,x;s,u_{0})dx\leq M_{1}^{*}\quad\forall\,t>s.$ (4.9)

By (1.10), (3.3) with $\tau_{0}=0$, (4.6), and (4.9),

$\displaystyle{\int_{\Omega}u^{q}(t,x;s,u_{0})dx}$
$\displaystyle{\leq\max\left\\{\int_{\Omega}u^{q}(\tau,x;s,u_{0})dx,M_{2}^{*}\right\\}\quad\forall\,t>\tau>s.}$

Letting $\tau\to s^{+}$, we have

$\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq M_{2}^{*}\quad\forall\,t\geq s.$ (4.10)

It then follows from (4.8), (4.9) and (4.10) that the set $\mathcal{E}$ is
invariant.

Next, we prove (1.14) holds. To this end, for any given
$0<\theta<1-\frac{2N}{q}$, choose $\beta\in(0,\frac{1}{2})$ such that

$2\beta-\frac{2N}{q}>\theta.$

By Lemma 2.5, there is $K_{1}$ such that for any $u_{0}\in\mathcal{E}$ and
$s\in\mathbb{R}$,

$\|u(t,\cdot;s,u_{0})\|_{C^{\theta}(\bar{\Omega})}\leq
K_{1}\|u(t,\cdot;s,u_{0})\|_{X^{\beta}_{q/2}}\quad\forall\,t>s.$ (4.11)

Note that for any $u_{0}\in\mathcal{E}$ and $s\in\mathbb{R}$,

$\displaystyle\|u(t,\cdot;s,u_{0})\|_{X^{\beta}_{q/2}}$
$\displaystyle\leq\|A^{\beta}e^{-A(t-s)}u_{0}\|_{L^{q/2}(\Omega)}$
$\displaystyle\quad+\chi\int_{s}^{t}\left\|A^{\beta}e^{-A(t-\lambda)}\nabla\cdot\left(\frac{u(\lambda,\cdot;s,u_{0})}{v(\lambda,\cdot;s,u_{0})}\nabla
v(\lambda,\cdot;s,u_{0})\right)\right\|_{L^{q/2}(\Omega)}d\lambda$
$\displaystyle\quad+\int_{s}^{t}\|A^{\beta}e^{-A(t-\lambda)}\big{(}(1+a(\lambda,\cdot))u(\lambda,\cdot;s,u_{0})-b(\lambda)u^{2}(\lambda,\cdot;s,u_{0})\big{)}\|_{L^{q/2}(\Omega)}d\lambda,$
(4.12)

where $A=A_{q/2}$. By Lemma 2.5 again, there is $K_{2}$ such that

$\|A^{\beta}e^{-A(t-s)}u_{0}\|_{L^{q/2}(\Omega)}\leq
K_{2}(t-s)^{-\beta(t-s)}e^{-\gamma(t-s)}\|u_{0}\|_{L^{q/2}}\quad\forall\,t>s.$
(4.13)

By Lemma 2.6, there is $K_{3}$ such that

$\displaystyle\chi\int_{s}^{t}\left\|A^{\beta}e^{-A(t-\lambda)}\nabla\cdot\left(\frac{u(\lambda,\cdot;s,u_{0})}{v(\lambda,\cdot;s,u_{0})}\nabla
v(\lambda,\cdot;s,u_{0})\right)\right\|_{L^{q/2}(\Omega)}d\lambda$
$\displaystyle\leq
K_{3}\chi\int_{s}^{t}(t-\lambda)^{-\beta-\frac{1}{2}-\epsilon}e^{-\gamma(t-\lambda)}\left\|\frac{u(\lambda,\cdot;s,u_{0})}{v(\lambda,\cdot;s,u_{0})}\cdot\nabla
v(\lambda,\cdot;s,u_{0})\right\|_{L^{q/2}(\Omega)}d\lambda$
$\displaystyle\leq\frac{K_{3}\chi}{\tilde{M}_{1}^{*}}\int_{s}^{t}(t-\lambda)^{-\beta-\frac{1}{2}-\epsilon}e^{-\gamma(t-\lambda)}\left\|{u(\lambda,\cdot;s,u_{0})}\cdot\nabla
v(\lambda,\cdot;s,u_{0})\right\|_{L^{q/2}(\Omega)}d\lambda\quad\text{(by Lemma
\ref{prelim-lm-2})}$
$\displaystyle\leq\frac{K_{3}\chi}{\tilde{M}_{1}^{*}}\int_{s}^{t}(t-\lambda)^{-\beta-\frac{1}{2}-\epsilon}e^{-\gamma(t-\lambda)}\left\|{u(\lambda,\cdot;s,u_{0})}\right\|_{L^{q}(\Omega)}\left\|\nabla
v(\lambda,\cdot;s,u_{0})\right\|_{L^{q}(\Omega)}d\lambda\quad\text{(by
H\"{o}lder inequality)}$
$\displaystyle\leq\frac{CK_{3}\chi(M_{2}^{*})^{2/q}}{\tilde{M}_{1}^{*}}\int_{s}^{t}(t-\lambda)^{-\beta-\frac{1}{2}-\epsilon}e^{-\gamma(t-\lambda)}d\lambda\quad\text{(by
Lemma \ref{prelim-lm-000} and Theorem \ref{main-thm1})}.$ (4.14)

By Theorem 1.1, there is $K_{4}>0$ such that

$\displaystyle\int_{s}^{t}\|A^{\beta}e^{-A(t-\lambda)}\big{(}(1+a(\lambda,\cdot))u(\lambda,\cdot;s,u_{0})-b(\lambda)u^{2}(\lambda,\cdot;s,u_{0})\big{)}\|_{L^{q/2}(\Omega)}d\lambda$
$\displaystyle\leq\int_{s}^{t}(t-\lambda)^{-\beta}e^{-\gamma(t-\lambda)}\big{(}|\Omega|^{2/q}+a_{\max}\|u(\lambda,\cdot;su_{0})\|_{L^{q/2}}+b_{\max}\|u^{2}(\lambda,\cdot;s,u_{0})\|_{L^{q/2}}\big{)}d\lambda$
$\displaystyle\leq
K_{4}\int_{s}^{t}(t-\lambda)^{-\beta}e^{-\gamma(t-\lambda)}d\lambda.$ (4.15)

By (4.11)-(4), for any $\tau>0$ and ${0<\theta<1-\frac{2N}{q}}$, there is
$M_{3}^{*}(\theta,\tau)>0$ such that (1.14) holds. (1) is thus proved.

(2) First of all, it is not difficult to see that (1.15) follows directly from
Lemma 3.1 and Theorem 1.1. It then suffices to prove (1.16).

For any $u_{0}$ satisfying ${(1.2)^{\prime}}$, by (1.15), there is
$T_{0}(u_{0})>0$ such that for any $s\in\mathbb{R}$,

$\int_{\Omega}u^{-p}(t,x;s,u_{0})dx\leq 2M_{1}^{*}\quad{\rm
and}\quad\int_{\Omega}u^{q}(t,x;s,u_{0})dx\leq 2M_{2}^{*}\quad\forall\,t\geq
s+T_{0}(u_{0}).$

Observe that

$(u(t,x;s,u_{0}),v(t,x;s,u_{0}))=(u(t,x;\tilde{s},\tilde{u}_{0}),v(t,x;\tilde{s},\tilde{u}_{0}))\quad\forall\,t\geq\tilde{s},$
(4.16)

where $\tilde{s}=s+T_{0}(u_{0})$ and
$\tilde{u}_{0}(x)=u(s+T_{0}(u_{0}),x;s,u_{0})$. Hence

$\int_{\Omega}u^{-p}(t,x;\tilde{s},\tilde{u}_{0})dx\leq 2M_{1}^{*}\quad{\rm
and}\quad\int_{\Omega}u^{q}(t,x;\tilde{s},\tilde{u}_{0})dx\leq
2M_{2}^{*}\quad\forall\,t\geq\tilde{s}.$

By the arguments in the proof of (1.14), there is $M_{4}^{*}(\theta)$ such
that

$\|u(t,\cdot;\tilde{s},\tilde{u}_{0})\|_{C^{\theta}(\bar{\Omega})}\leq
M_{4}^{*}(\theta)\quad\forall\,t\geq\tilde{s}+1.$

This together with (4.16) implies that (1.16) holds with
$T(u_{0})=T_{0}(u_{0})+1$. ∎

We now prove Theorem 1.3. In the rest of this section, to indicate the
dependence of $(u(t,x;s,u_{0}),v(t,x;s,u_{0}))$ on $a(t,x),b(t,x)$, we put

$(u(t,x;s,u_{0},a,b),v(t,x;s,u_{0},a,b))=(u(t,x;s,u_{0}),v(t,x;s,u_{0})).$

###### Proof of Theorem 1.3.

We prove this theorem by contradiction. Assume that there is no $m^{*}>0$ such
that (1.18) holds. Then for any $n\in\mathbb{N}$, there is $u_{n}$ satisfying
${(1.2)^{\prime}}$ such that

$\liminf_{t-s\to\infty}\inf_{x\in\Omega}u(t,x;s,u_{n},a,b)\leq
m_{n}:=\frac{1}{n}.$ (4.17)

Let $p,q,\theta,M_{1}^{*},M_{2}^{*}$, and $M_{4}^{*}(\theta)$ be as in Theorem
1.2. By Theorem 1.2 and Lemma 4.1, there is $T_{n}>0$ such that

$\|u(t,\cdot;s,u_{n},a,b)\|_{C^{\theta}(\bar{\Omega})}\leq
M_{4}^{*}(\theta)\quad\forall\,t-s\geq T_{n}$ (4.18)

and

$\int_{\Omega}u(t,x;s,u_{n},a,b)dx\geq\frac{\tilde{M}_{1}^{*}}{2}\quad\forall\,t-s\geq
T_{n}.$ (4.19)

By (4.17), there are $t_{n},s_{n}\in\mathbb{R}$ with $t_{n}-s_{n}\geq T_{n}+1$
and $x_{n}\in\bar{\Omega}$ such that

$u(t_{n},x_{n};s_{n},u_{n},a,b)\leq\frac{2}{n}.$ (4.20)

Let

$a_{n}(t,x)=a(t+t_{n}-1,x),\quad b_{n}(t,x)=b(t+t_{n}-1,x).$

Observe that

$\displaystyle u(t_{n},x;s,u_{n},a,b)$
$\displaystyle=u(t_{n},x;t_{n}-1,u(t_{n}-1,\cdot;s_{n},u_{n},a,b),a,b)$
$\displaystyle=u(1,x;0,u(t_{n}-1,\cdot;s_{n},u_{n},a,b),a_{n},b_{n}).$ (4.21)

By (4.18) and (H), without loss of generality, we may assume that there are
$u_{0}^{*},u_{1}^{*}\in C(\bar{\Omega})$ and $a^{*}(t,x),b^{*}(t,x)$
satisfying (H) such that

$\lim_{n\to\infty}u(t_{n}-1,x;s_{n},u_{n},a,b)=u_{0}^{*}(x),\quad\lim_{n\to\infty}u(t_{n},x;s_{n},u_{n},a,b)=u_{1}^{*}(x)$

uniformly in $x\in\bar{\Omega}$, and

$\lim_{n\to\infty}a_{n}(t,x)=a^{*}(t,x),\quad\lim_{n\to\infty}b_{n}(t,x)=b^{*}(t,x)$

uniformly in $x\in\bar{\Omega}$ and locally uniformly in $t\in\mathbb{R}$. It
then follows from Lemma 2.8 and (4) that

$u_{1}^{*}(x)=u(1,x;0,u_{0}^{*},a^{*},b^{*}).$ (4.22)

By (4.18), (4.19), and the Dominated Convergence Theorem,

$\int_{\Omega}u_{0}^{*}(x)=\lim_{n\to\infty}\int_{\Omega}u(t_{n}-1,x;s_{n},u_{n},a,b)dx\geq\frac{\tilde{M}_{1}^{*}}{2}.$

Then by (4.22) and the strong maximum principle for parabolic equations,

$\inf_{x\in\bar{\Omega}}u_{1}^{*}(x)>0.$

This together with (4.20) implies that

$\frac{2}{n}\geq\inf_{x\in\Omega}u(t_{n},x;s_{n},u_{n},a,b)\geq\frac{1}{2}\inf_{x\in\Omega}u_{1}^{*}(x)\quad\forall\,n\gg
1,$

which is a contradiction. Therefore, there is $m^{*}>0$ such that for any
$u_{0}$ satisfying (1.2),

$\liminf_{t-s\to\infty}\inf_{x\in\Omega}u(t,x;s,u_{0},a,b)\geq m^{*}.$

The theorem is thus proved. ∎

##  5 Existence of positive entire solutions

In this section, we study the existence of positive entire solutions and prove
Theorem 1.4.

We first prove a lemma.

######  Lemma 5.1.

Let $p>0$, $q>2N$, and $M_{1}^{*},M_{2}^{*}>0$ be as in Theorem 1.2. The set

$\mathcal{E}=\left\\{u\in C^{0}(\bar{\Omega})\,|\,u\geq
0,\,\,\int_{\Omega}u(x)\leq M_{0}^{*},\,\,\int_{\Omega}u^{-p}(x)dx\leq
M_{1}^{*},\,\,\int_{\Omega}u^{q}(x)\leq M_{2}^{*}\right\\}$

is a convex, closed subset of $C^{0}(\bar{\Omega})$.

###### Proof.

First, we prove the set $\mathcal{E}$ is convex. Note that $g(u)=u^{r}$ is a
convex function on $(0,\infty)$, where $r=1$, $-p$ or $q$. Hence, for any
$\lambda\in[0,1]$ and $u_{*},u^{*}>0$, we have

$g(\lambda u_{*}+(1-\lambda)u^{*})\leq\lambda g(u_{*})+(1-\lambda)g(u^{*}).$
(5.1)

For any $u_{1},u_{2}\in\mathcal{E},$ we have $\int_{\Omega}u_{1}^{-p}(x)dx\leq
M_{1}^{*}$, $\int_{\Omega}u_{2}{{}^{-p}}(x)dx\leq M_{1}^{*}$, and
$u_{1}(x)>0,u_{2}(x)>0$ for a.e. $x\in\Omega$. By (5.1), we have that

$g(\lambda u_{1}(x)+(1-\lambda)u_{2}(x))\leq\lambda
g(u_{1}(x))+(1-\lambda)g(u_{2}(x))\quad{\rm for}\quad a.\,e.\,\,\,x\in\Omega.$

This implies that

$\displaystyle\int_{\Omega}g(\lambda u_{1}(x)+(1-\lambda)u_{2}(x))dx$
$\displaystyle\leq\lambda\int_{\Omega}g(u_{1}(x))dx+(1-\lambda)\int_{\Omega}g(u_{2}(x))dx.$

This implies that for any $\lambda\in[0,1]$ and $u_{1},u_{2}\in\mathcal{E}$,
$\lambda u_{1}+(1-\lambda)u_{2}\in\mathcal{E}.$ Thus, the set $\mathcal{E}$ is
convex.

Next, we prove the set $\mathcal{E}$ is closed. Suppose that
$u_{n}\in\mathcal{E}$, $u\in C^{0}(\bar{\Omega})$, and $u_{n}\rightarrow u$ in
$C^{0}(\bar{\Omega})$. Then $u(x)\geq 0$ for any $x\in\bar{\Omega}$, and for
any $\varepsilon>0$, there is $N_{\varepsilon}$ such that for any $n\geq
N_{\varepsilon}$, there holds

$|u_{n}(x)-u(x)|<\varepsilon\quad\forall\,x\in\bar{\Omega}.$

This implies that

$0\leq u_{n}(x)=u_{n}(x)-u(x)+u(x)\leq\varepsilon+u(x)\quad\forall\,n\geq
N_{\varepsilon},\,\,x\in\bar{\Omega}.$ (5.2)

For any $\varepsilon>0$, let

$\Omega_{\varepsilon}=\\{x\in\Omega\,|\,u(x)\leq\varepsilon\\}.$

Let

$\Omega_{0}=\\{x\in\Omega\,|\,u(x)=0\\}.$

By (5.2),

$0\leq u_{n}(x)\leq 2\varepsilon\quad\forall\,x\in\Omega_{\varepsilon}.$

Since $\int_{\Omega}u_{n}^{-p}(x)dx\leq M_{1}^{*}$, we have that

$|\Omega_{\epsilon}|=\int_{\Omega}u_{n}^{-p}\cdot
u_{n}^{p}\leq\int_{\Omega}u_{n}^{-p}\cdot(2\epsilon)^{p}\leq(2\epsilon)^{p}\int_{\Omega}u_{n}^{-p}\leq(2\epsilon)^{p}M_{1}^{*}.$

This implies that

$|\Omega_{0}|=0.$

Let $g(u)$ be as in the above. Then by Fatou’s lemma, we have

$\displaystyle\int_{\Omega}g(u(x))dx=\int_{\Omega\setminus\Omega_{0}}g(u(x))dx=\int_{\Omega\setminus\Omega_{0}}\lim_{n\rightarrow\infty}g(u_{n}(x))dx\leq\lim_{n\rightarrow\infty}\inf\int_{\Omega\setminus\Omega_{0}}g(u_{n}(x))dx.$

This implies that

$\int_{\Omega}u(x)dx\leq M_{0}^{*},\,\,\int_{\Omega}u^{-p}(x)dx\leq
M_{1}^{*},\,\,\int_{\Omega}u^{q}(x)dx\leq{M_{2}^{*}}.$

Hence $u\in\mathcal{E}$ and the set $\mathcal{E}$ is closed. ∎

We now prove Theorem 1.4.

###### Proof of Theorem 1.4.

(1) First of all, fix $u_{0}\in\mathcal{E},$ where $\mathcal{E}$ as in (1.13).
By Theorem 1.2, $u(t,\cdot;s,u_{0})\in\mathcal{E}$ for any $t>s$.

Next, for any $n\in\mathbb{N}$, define $u_{n}(x)=u(0,x,;-n,u_{0})$. Fix
$\theta\in(0,\frac{2N}{q})$. By Theorem 1.2 again, there is
$\tilde{M}_{3}^{*}>0$ such that

$\|u(t,\cdot;-n,u_{n})\|_{C^{\theta}}=\sup_{x\in\Omega}{|u(t,x;-n,u_{0})|}+\sup{\frac{|u(t,x;-n,u_{0})-u(t,y;-n,u_{0})|}{|x-y|^{\theta}}}\leq\tilde{M}_{3}^{*}$
(5.3)

for all $t\geq-n+1$ and $n\geq 1$. This implies that the sequence
$\\{u_{n}(\cdot)\\}$ is uniformly bounded and equicontinuous on
$\bar{\Omega}$. By the Arzelá-Ascoli Theorem and Lemma 5.1, there are $n_{k}$,
$u^{*}_{0}\in\mathcal{E}$ such that $u_{n_{k}}\rightarrow u^{*}_{0}$ in
$C^{0}(\bar{\Omega})$ as $n_{k}\to\infty.$

Note that

$u(\cdot,t;-n_{k},u_{0})=u(\cdot,t;0,u(\cdot,0;-n_{k},u_{0}))=u(\cdot,t;0,u_{n_{k}})\quad\forall\,t>0.$

By Lemma 2.8, for any $t>0$,

$\lim_{n_{k}\to\infty}u(\cdot,t;-n_{k},u_{0})=\lim_{n_{k}\to\infty}u(\cdot,t;0,u_{n_{k}})=u(\cdot,t;0,u^{*}_{0})\quad{\rm
in}\quad C^{0}(\bar{\Omega}).$

Since $v(\cdot,t;0,u^{*}_{0})=A^{-1}u(\cdot,t;0,u^{*}_{0}),$ we also have that
for any $t>0$,

$\lim_{n_{k}\to\infty}v(\cdot,t;-n_{k},u_{0})=\lim_{n_{k}\to\infty}A^{-1}u(\cdot,t;-n_{k},u_{0})=v(\cdot,t;0,u^{*}_{0})\quad{\rm
in}\quad C^{0}(\bar{\Omega}).$

We now prove that $u(\cdot,t;0,u^{*}_{0})$ has a backward extension on
$(-\infty,0)$. By (5.3) and the Arzelá-Ascoli Theorem, without loss of
generality, we may assume that for any $m\in\mathbb{N}$, there is
$u_{m}^{*}(\cdot)\in\mathcal{E}$ such that $u(\cdot,-m;-n_{k},u_{0})\to
u_{m}^{*}(\cdot)$ in $C^{0}(\bar{\Omega})$. By Lemma 2.8 again,

$\lim_{n_{k}\to\infty}u(\cdot,t;-n_{k},u_{0})=\lim_{n_{k}\to\infty}u(\cdot,t;-m,u(\cdot,-m;-n_{k},u_{0}))=u(\cdot,t;-m,u_{m}^{*})$

and for $t>-m$. Note that $u_{0}^{*}=u(\cdot,0;-m,u_{m}^{*})$. This implies
that $u^{*}(x,t;0,u_{0}^{*})$ has a backward extension up to $t=-m$. Letting
$m\to\infty$ yields that $u^{*}(x,t)$ has a backward extension on
$(-\infty,0)$.

Finally, let us denote $u^{*}(t,x)=u^{*}(t,x;0,u_{0}^{*}),$ hence
$v^{*}(t,x)=A^{-1}u^{*}(t,x).$ Then $(u^{*}(t,x),v^{*}(t,x))$ is an entire
solution of (1.1) and $u^{*}(t,\cdot)\in\mathcal{E}$ for any $t\in\mathbb{R}$.
(1) is thus proved.

(2) Assume that there exists $T>0$ such that $a(t+T,x)=a(t,x)$ and
$b(t+T,x)=b(t,x)$. First, define the map
$\mathcal{T}(T):\mathcal{E}\rightarrow C^{0}(\bar{\Omega})$ as follows:

$\mathcal{T}(T)u_{0}=u(T,\cdot;0,u_{0})\quad\forall\,u_{0}\in\mathcal{E}.$

By Theorem 1.2, $\mathcal{T}(T)$ maps $\mathcal{E}$ into $\mathcal{E}$. By
Lemma 2.7 and Theorem 1.2, $\mathcal{T}(T):\mathcal{E}\to\mathcal{E}$ is
continuous and relatively compact in $C^{0}(\bar{\Omega})$. By Lemma 5.1,
$\mathcal{E}$ is a convex closed subset of $C^{0}(\bar{\Omega})$. Then, by
Schauder fixed point theorem (see Lemma 2.2), one can find
$u^{T}\in\mathcal{E}$ such that $\mathcal{T}(T)u^{T}=u^{T},$ that is,
$u(\cdot,T;s,u^{T})=u^{T}(\cdot).$ Thus,

$u(t+T,\cdot;0,u^{T})=u(t,\cdot;T,u(T,\cdot;0,u^{T}))=u(t,\cdot;0,u^{T}).$

Hence, $u(t,\cdot;0,u^{T})$ is periodic with period $T$. Moreover, the second
equation of (1.3) and the uniqueness of solutions of

$\begin{cases}-\Delta v+v=u(t,\cdot;0,u^{T}),&x\in\Omega\cr\frac{\partial
v}{\partial n}=0,\quad x\in\partial\Omega,\end{cases}$

it is obtained that $v(t,\cdot;0,u^{T})=(I-\Delta)^{-1}u(t,\cdot;0,u^{T})$ is
periodic with period $T,$ which concludes that
$(u(t,\cdot;0,u^{T}),v(t,\cdot;0,u^{T}))$ is a positive periodic solution of
(1.3) with $u(t,\cdot;0,u^{T})\in\mathcal{E}$ for any $t\in\mathbb{R}$.. (2)
is thus proved.

(3) Assume that $a(t,x)=a(x)$ and $b(t,x)=b(x).$ Observe that every $T>0$ is a
period for $a(t,x)$ and $b(t,x).$ By (2), for fixed $T>0,$ there exists
$u^{T}\in\mathcal{E}$ such that
$(u(t,\cdot;0,u^{T}),(I-\Delta)^{-1}u(t,\cdot;0,u^{T}))$ is a positive
periodic solution of (1.1) with period $T$.

Let $T_{n}=\frac{1}{n}$ and $u_{n}(x)=u^{T_{n}}(x)$. Then
$u_{n}(x)=u(kT_{n},x;0,u_{n})$ for all $k\in\mathbb{N}$. By Theorem 1.2,
$\\{u_{n}(\cdot)\\}$ is a uniformly bounded and equi-continuous sequence in
$C^{0}(\bar{\Omega})$. Then by the Arzelá-Ascoli theorem, there exist a
subsequence $\\{u_{n_{k}}\\}$ and $u^{*}\in\mathcal{E}$ such that
$u_{n_{k}}\to u^{*}$ in $C^{0}(\bar{\Omega})$ as $n_{k}\to\infty.$

We claim that

$u(\cdot,t;0,u^{*})=u^{*}(\cdot)\quad\text{for all}\,\,t\geq 0.$ (5.4)

For any $t>0$ and $\epsilon>0$, let $\tau_{n_{k}}\in[0,T_{n_{k}})$ be such
that

$\frac{t-\tau_{n_{k}}}{T_{n_{k}}}\in\mathbb{Z}^{+}.$

Then

$\tau_{n_{k}}\to 0\quad{\rm and}\quad k\to\infty.$

By Lemma 2.7,

$\|u(\tau_{n_{k}},\cdot;0,u^{*})-u^{*}(\cdot)\|_{\infty}<\epsilon\quad\forall\,k\gg
1.$

By Lemma 2.8,

$\|u(\tau,\cdot;0,u^{*})-u(\tau,\cdot;0,u_{n_{k}})\|_{\infty}<\epsilon\quad\forall\,\tau\in[0,t]\quad{\rm
and}\quad\forall\,k\gg 1.$

It then follows that

$\displaystyle|u(t,x;0,u^{*})-u^{*}(x)|$
$\displaystyle\leq|u(t,x;0,u^{*})-u(t,x;0,u_{n_{k}})|+|u(t,x;0,u_{n_{k}})-u^{*}(x)|\quad(\text{choose}\,\,k\gg
1)$
$\displaystyle=|u(t,x;0,u^{*})-u(t,x;0,u_{n_{k}})|+|u(\tau_{n_{k}},x;0,u_{n_{k}})-u^{*}(x)|$
$\displaystyle\leq|u(t,x;0,u^{*})-u(t,x;0,u_{n_{k}})|+|u(\tau_{n_{k}},x;0,u_{n_{k}})-u(\tau_{n_{k}},x;0,u^{*})|$
$\displaystyle\quad+|u(\tau_{n_{k}},x;0,u^{*})-u^{*}(x)|$ $\displaystyle\leq
3\epsilon\quad\forall\,x\in\bar{\Omega}.$

Letting $\epsilon\to 0$ entails (5.4). This shows that the solution pair
$(u(\cdot,t;0,u^{*}),v(\cdot,t;0,u^{*}))$ with
$v(\cdot,t;0,u^{*})=(I-\Delta)^{-1}u(\cdot,t;0,u^{*})$ is a steady state
solution of (1.1) with $u^{*}(\cdot)\in\mathcal{E}$. (3) is thus proved. ∎

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