# Two-species chemotaxis-competition system with singular sensitivity: Global
existence, boundedness, and persistence

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

###### Abstract

This paper is concerned with the following parabolic-parabolic-elliptic
chemotaxis system with singular sensitivity and Lotka-Volterra competitive
kinetics,

$\begin{cases}u_{t}=\Delta u-\chi_{1}\nabla\cdot(\frac{u}{w}\nabla
w)+u(a_{1}-b_{1}u-c_{1}v),\quad&x\in\Omega\cr v_{t}=\Delta
v-\chi_{2}\nabla\cdot(\frac{v}{w}\nabla
w)+v(a_{2}-b_{2}v-c_{2}u),\quad&x\in\Omega\cr 0=\Delta w-\mu w+\nu u+\lambda
v,\quad&x\in\Omega\cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial
n}=\frac{\partial w}{\partial n}=0,\quad&x\in\partial\Omega,\end{cases}$ (0.1)

where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain, and
$\chi_{i}$, $a_{i}$, $b_{i}$, $c_{i}$ ($i=1,2$) and $\mu,\,\nu,\,\lambda$ are
positive constants. This is the first work on two-species chemotaxis-
competition system with singular sensitivity and Lotka-Volterra competitive
kinetics. Among others, we prove that for any given nonnegative initial data
$u_{0},v_{0}\in C^{0}(\bar{\Omega})$ with $u_{0}+v_{0}\not\equiv 0$, (0.1) has
a unique globally defined classical solution
$(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0}))$ with
$u(0,x;u_{0},v_{0})=u_{0}(x)$ and $v(0,x;u_{0},v_{0})=v_{0}(x)$ in any space
dimensional setting with any positive constants $\chi_{i},a_{i},b_{i},c_{i}$
($i=1,2$) and $\mu,\nu,\lambda$. Moreover, we prove that there is
$\chi^{*}(\mu,\chi_{1},\chi_{2})>0$ satisfying

$\displaystyle\chi^{*}(\mu,\chi_{1},\chi_{2})=\begin{cases}\frac{\mu\chi^{2}}{4}&{\rm
if}\,\,0<\chi<2\cr\mu(\chi-1)&{\rm if}\,\,\chi\geq 2,\end{cases}\quad\quad{\rm
when\,\chi_{1}=\chi_{2}:=\chi}$ and
$\displaystyle\chi^{*}(\mu,\chi_{1},\chi_{2})\leq\min\Big{\\{}\mu\chi_{2}+\frac{\mu(\chi_{1}-\chi_{2})^{2}}{4\chi_{2}},\mu\chi_{1}+\frac{\mu(\chi_{2}-\chi_{1})^{2}}{4\chi_{1}}\Big{\\}},\quad\quad{\rm
when\,\chi_{1}\not=\chi_{2}}$

such that the condition

$\min\\{a_{1},a_{2}\\}>\chi^{*}(\mu,\chi_{1},\chi_{2})$

implies

$\limsup_{t\to\infty}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{\infty}\leq
M^{*}\,\,{\rm
and}\,\,\liminf_{t\to\infty}\inf_{x\in\Omega}(u(t,x,u_{0},v_{0})+v(t,x;u_{0},v_{0}))\geq
m^{*}$

for some positive constants $M^{*},m^{*}$ independent of $u_{0},v_{0}$, the
latter is referred to as combined pointwise persistence.

Key words. Parabolic-parabolic-elliptic chemotaxis system, singular
sensitivity, Lotka-Volterra competitive kinetics, global existence, global
boundedness, combined mass persistence, combined pointwise persistence.

2020 Mathematics subject Classification. 35K51, 35K57, 35M33, 35Q92, 92C17,
92D25.

##  1 Introduction and Main Results

Chemotaxis refers to the movement of cells or organisms in response to
chemicals in their environments, and plays a crucial role in many biological
processes such as immune system response, tumor growth, population dynamics,
gravitational collapse, the governing of immune cell migration. Since the
pioneering works by Keller and Segel ([20], [21]) on chemotaxis models, a lot
of works have been carried out on the qualitative properties of various
chemotaxis models such as the analysis of global existence, boundedness, blow-
up in finite time, and asymptotic behavior of globally defined solutions, etc.
The reader is referred to [3, 12, 13] and the references therein for some
detailed introduction into the mathematics of chemotaxis models.

There are a large number of works on various two competing species chemotaxis
models. For example, consider the following two-species chemotaxis system,

$\begin{cases}u_{t}=\Delta u-\nabla\cdot(u\chi_{1}(w)\nabla
w)+u(a_{1}-b_{1}u-c_{1}v),\quad&x\in\Omega\cr v_{t}=\Delta
v-\nabla\cdot(v\chi_{2}(w)\nabla
w)+v(a_{2}-b_{2}v-c_{2}u),\quad&x\in\Omega\cr\tau w_{t}=\Delta w-{\mu w}+\nu
u+\lambda v,\quad&x\in\Omega\cr\frac{\partial u}{\partial n}=\frac{\partial
v}{\partial n}=\frac{\partial w}{\partial
n}=0,\quad&x\in\partial\Omega,\end{cases}\,$ (1.1)

where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain,
$a_{1},b_{1},c_{1},a_{2},b_{2},c_{2},\nu,\lambda$ are positive numbers, and
$\tau\geq 0$. Biologically, (1.1) models the evolution of two competitive
species subject to a chemical substance, which is produced by the two species
themselves. Here the unknown functions $u(t,x)$ and $v(t,x)$ represent the
population densities of two competitive biological species and $w(t,x)$
represents the concentration of the chemical substance. The terms
$u(a_{1}-b_{1}u-c_{1}v)$ and $v(a_{2}-b_{2}v-c_{2}u)$ are referred to as
Lotka-Volterra competitive terms. The parameter $\mu$ is the degradation rate
of the chemical substance and $\nu$ and $\lambda$ are the production rates of
the chemical substance by the species $u$ and $v$, respectively. $\tau\geq 0$
is related to the diffusion rate of the chemical substance. The functions
$\chi_{1}(w)$ and $\chi_{2}(w)$ reflect the strength of the chemical substance
on the movements of two species, and are referred to as chemotaxis sensitivity
functions or coefficients.

When $\chi_{1}(w)\equiv\chi_{1}>0$, $\chi_{2}(w)\equiv\chi_{2}>0$, and
$\tau=0$, it is known that if $N\leq 2$, or $N\geq 3$ and $\chi_{i}$ is small
relative to $b_{i}$ and $c_{i}$ ($i=1,2$), then for any nonnegative initial
data $u_{0},v_{0}\in C^{0}(\bar{\Omega})$, system (1.1) possesses a unique
globally defined classical solution
$(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0}))$ with
$u(0,x;u_{0},v_{0})=u_{0}(x)$ and $v(0,x;u_{0},v_{0})=v_{0}(x)$ (see [16, 19,
24, 29, 30] and references therein). Moreover, the large time behaviors of
globally defined classical solutions of (1.1) such as competitive exclusion,
coexistence, stabilization, etc., are investigated in [6, 16, 18, 19, 25, 26,
29, 30], etc.

When $\chi_{1}(\omega)\equiv\chi_{1}>0$, $\chi_{2}(w)\equiv\chi_{2}>0$ and
$\tau=1$, it is proved that, if $N\leq 2,$ or $N\geq 3$ and $\chi_{1}$ and
$\chi_{2}$ are small relative to other parameters in (1.1), then for any
nonnegative initial data $u_{0},v_{0}\in C(\bar{\Omega})$, $w_{0}\in
W^{1,\infty}(\Omega)$, system (1.1) possesses a unique globally defined
classical solution
$(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0}))$ with
$u(0,x;u_{0},v_{0})=u_{0}(x)$, $v(0,x;u_{0},v_{0})=v_{0}(x)$, and
$w(0,x;u_{0},v_{0})=w_{0}(x)$ (see [2, 24, 31], etc.). Moreover, the large-
time behaviors of globally defined classical solutions are investigated in [2,
31], etc. We refer to the readers to the articles [14, 31] for the further
details.

The aim of current paper is to investigate the global existence, boundedness,
and combined persistence of classical solutions of (1.1) with
$\chi_{1}(w)=\frac{\chi_{1}}{w}$ and $\chi_{2}(w)=\frac{\chi_{2}}{w}$ for some
positive constants $\chi_{1}$ and $\chi_{2}$, and $\tau=0$, that is, the
following parabolic-parabolic-elliptic chemotaxis system with singular
sensitivity and Lotka-Volterra competitive kinetics,

$\begin{cases}u_{t}=\Delta u-\chi_{1}\nabla\cdot(\frac{u}{w}\nabla
w)+u(a_{1}-b_{1}u-c_{1}v),\quad&x\in\Omega\cr v_{t}=\Delta
v-\chi_{2}\nabla\cdot(\frac{v}{w}\nabla
w)+v(a_{2}-b_{2}v-c_{2}u),\quad&x\in\Omega\cr 0=\Delta w-\mu w+\nu u+\lambda
v,\quad&x\in\Omega\cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial
n}=\frac{\partial w}{\partial n}=0,\quad&x\in\partial\Omega.\end{cases}$ (1.2)

It is seen that the chemotaxis sensitivities $\frac{\chi_{i}}{w}$ ($i=1,2$)
are singular near $w=0$, reflecting an inhibition of chemotactic migration at
high signal concentrations. Such a sensitivity describing the living
organisms’ response to the chemical signal was derived by the Weber-Fechner
law (see [21]).

We consider classical solutions of (1.2) with initial functions
$u_{0},v_{0}\in C^{0}(\bar{\Omega})$ with $u_{0}\geq 0$, $v_{0}\geq 0$, and
$\int_{\Omega}(u_{0}+v_{0})>0$. Note that for such initial functions, if
$v_{0}=0$ (resp. $u_{0}=0$), then $v(t,x)\equiv 0$ (resp. $u(t,x)\equiv 0$) on
the existence interval. Note also that if $v(t,x)\equiv 0$, then (1.2) becomes

$\begin{cases}u_{t}=\Delta u-\chi_{1}\nabla\cdot(\frac{u}{w}\nabla
w)+u(a_{1}-b_{1}u),\quad&x\in\Omega\cr 0=\Delta w-\mu w+\nu
u,\quad&x\in\Omega\cr\frac{\partial u}{\partial n}=\frac{\partial w}{\partial
n}=0,\quad&x\in\partial\Omega,\end{cases}$ (1.3)

and if $u(t,x)\equiv 0$, then (1.2) becomes

$\begin{cases}v_{t}=\Delta v-\chi_{2}\nabla\cdot(\frac{v}{w}\nabla
w)+v(a_{2}-b_{2}v),\quad&x\in\Omega\cr 0=\Delta w-\mu w+\lambda
v,\quad&x\in\Omega\cr\frac{\partial v}{\partial n}=\frac{\partial w}{\partial
n}=0,\quad&x\in\partial\Omega.\end{cases}\,$ (1.4)

Systems (1.3) and (1.4) are essentially the same, and are referred to as one
species chemotaxis models with logistic source and singular sensitivity. They
have been studied in many works (see [4, 5, 7, 9, 10, 22, 23, 27], etc.). Let
us briefly review some known results for one species chemotaxis models with
logistic source and singular sensitivity.

Consider (1.3) with $a_{1}=b_{1}\equiv 0$ and $\mu=\nu=1$. Fujie, Winkler, and
Yokota in [10] proved the global existence and boundedness of positive
classical solutions when $\chi_{1}<\frac{2}{N}$ and $N\geq 2$. More recently,
Fujie and Senba in [8] proved the global existence and boundedness of
classical positive solutions for the case of $N=2$ for any $\chi_{1}>0$. The
existence of finite-time blow-up is then completely ruled out for any
$\chi_{1}>0$ in the case $N=2$. When $N\geq 3$, finite-time blow-up may occur
(see [27]).

Consider (1.3) with $a_{1},b_{1}>0$. Central questions include whether the
logistic source prevents the occurrence of finite-time blow-up in (1.3) (i.e.
any positive solution exists globally); if so, whether any globally defined
positive solution is bounded, and what is the long time behavior of globally
defined bounded positive solutions, etc. When $N=2$ and $a_{1},b_{1}$ are
positive constants, it is proved in [9] that finite-time blow-up does not
occur (see [9, Theorem 1.1]), and moreover, if

$a_{1}>\begin{cases}\frac{\mu\chi_{1}^{2}}{4},&\text{if $0<\chi_{1}\leq 2$}\\\
\mu(\chi_{1}-1),&\text{if $\chi_{1}>2$},\end{cases}$ (1.5)

then any globally defined positive solution is bounded. Under some additional
assumption, it is proved in [7] that the constant solution
$(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ is exponentially stable (see [7,
Theorem 1]). Very recently, among others, we proved that in any space
dimensional setting, logistic kinetics prevents the occurrence of finite-time
blow-up in (1.3) even for arbitrarily large $\chi_{1}$ (see [22, Theorem
1.2(1)]). Moreover, under the assumption (1.5), we proved that
$\int_{\Omega}u^{-q}$ for some $q>0$ and $\int_{\Omega}u^{p}$ for some $p>2N$
are bounded (see [23, Theorem 1.1]), and that globally defined positive
solutions are uniformly bounded and are away from $0$ (see [22, Theorem
1.1(2)] and [23, Theorem 1.3]).

We remark that in the recent paper [28], the following one-species two-
chemicals chemotaxis system is considered,

$\begin{cases}u_{t}=\Delta u-\chi_{1}\nabla\cdot(\frac{u}{v}\nabla
v)+u(a_{1}-b_{1}u),\quad&x\in\Omega\cr 0=\Delta
v-\alpha_{1}v+\beta_{1}w,\quad&x\in\Omega\cr 0=\Delta
w-\alpha_{2}w+\beta_{2}u,\quad&x\in\Omega\cr\frac{\partial u}{\partial
n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial
n}=0,\quad&x\in\partial\Omega,\end{cases}$ (1.6)

where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain,
$a_{1},b_{1},\alpha_{1},\alpha_{2},\beta_{2},\beta_{2}$ are positive numbers.
It is proved in [28] that, for any appropriately regular initial data, system
(1.6) has a unique globally defined classical solution for any $N\geq 1$ and
$\chi>0$.

However, as far as we know, there is little study on the two-species
chemotaxis system (1.2). It is the aim of this paper to investigate the global
existence, boundedness, and combined persistence of classical solutions of
(1.2).

######  Definition 1.1.

For given $u_{0}(\cdot)\in C^{0}(\bar{\Omega})$ and $v_{0}(\cdot)\in
C^{0}(\bar{\Omega})$ satisfying that $u_{0}\geq 0$, $v_{0}\geq 0$, and
$\int_{\Omega}(u_{0}(x)+v_{0}(x))dx>0$, we say $(u(t,x),v(t,x),w(t,x))$ is a
classical solution of (1.2) on $(0,T)$ for some $T\in(0,\infty]$ with initial
condition $(u(0,x),v(0,x))=(u_{0}(x),v_{0}(x))$ if

$u(\cdot,\cdot),v(\cdot,\cdot)\in C([0,T)\times\bar{\Omega})\cap
C^{1,2}((0,T)\times\bar{\Omega}),\quad w(\cdot,\cdot)\in
C^{0,2}((0,T)\times\bar{\Omega}),$ (1.7) $\lim_{t\to
0+}\|u(t,\cdot)-u_{0}(\cdot)\|_{C^{0}(\bar{\Omega})}=0,\quad\lim_{t\to
0+}\|v(t,\cdot)-v_{0}(\cdot)\|_{C^{0}(\bar{\Omega})}=0,$ (1.8)

and $(u(t,x),v(t,x),w(t,x))$ satisfies (1.2) for all
$(t,x)\in(0,T)\times\Omega$.

The following proposition on the local existence of classical solutions of
(1.2) can be proved by the similar arguments as those in [9, Lemma 2.2].

######  Proposition 1.1 (Local existence).

For given $u_{0}(\cdot)\in C^{0}(\bar{\Omega})$ and $v_{0}(\cdot)\in
C^{0}(\bar{\Omega})$ satisfying that $u_{0}\geq 0$, $v_{0}\geq 0$, and
$\int_{\Omega}(u_{0}(x)+v_{0}(x))dx>0$, there exists
$T_{\max}(u_{0},v_{0})\in(0,\infty]$ such that (1.2) possesses a unique
positive classical solution, denoted by $(u(t,x;u_{0},v_{0})$,
$v(t,x;u_{0},v_{0})$, $w(t,x;u_{0},v_{0}))$, on $(0,T_{\max}(u_{0},v_{0}))$
with initial condition
$(u(0,x;u_{0},v_{0}),v(0,x;u_{0},v_{0}))=(u_{0}(x),v_{0}(x))$. Moreover if
$T_{\max}(u_{0},v_{0})<\infty,$ then either

$\limsup_{t\nearrow
T_{\max}(u_{0},v_{0})}\left(\left\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\right\|_{C^{0}(\bar{\Omega})}\right)=\infty,$
(1.9)

or

$\liminf_{t\nearrow
T_{\max}(u_{0},v_{0})}\inf_{x\in\Omega}w(t,x;u_{0},v_{0})=0.$ (1.10)

We will focus on the following problems in this paper: whether
$(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0})$, $w(t,x;u_{0},v_{0}))$ exists
globally, i.e., $T_{\max}(u_{0},v_{0})=\infty$, for any $u_{0}(\cdot)\in
C^{0}(\bar{\Omega})$ and $v_{0}(\cdot)\in C^{0}(\bar{\Omega})$ satisfying that
$u_{0}\geq 0$, $v_{0}\geq 0$, and $\int_{\Omega}(u_{0}(x)+v_{0}(x))dx>0$; If
$T_{\max}(u_{0},v_{0})=\infty$, whether
$(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0})$, $w(t,x;u_{0},v_{0}))$ is bounded
above and stays away from $0$.

We point out that in the study of global existence and boundedness of
classical solutions of one species chemotaxis model (1.3) with singular
sensitivity, it is crucial to prove the boundedness of
$(\int_{\Omega}u(t,x)dx)^{-1}$ and the boundedness of
$\int_{\Omega}u^{p}(t,x)dx$ for some $p\gg 1$. The novel idea discovered in
this paper for the study of global existence and boundedness of classical
solutions of (1.2) is to prove the boundedness of
$(\int_{\Omega}(u(t,x)+v(t,x))dx)^{-1}$ and the boundedness of
$\int_{\Omega}(u(t,x)+v(t,x))^{p}dx$ for $p\gg 1$. Note that for (1.2),
$(\int_{\Omega}u(t,x)dx)^{-1}$ (resp. $(\int_{\Omega}v(t,x)dx)^{-1}$) may not
be bounded. Note also that the boundedness of
$(\int_{\Omega}(u(t,x)+v(t,x))dx)^{-1}$ is strongly related to the boundedness
of $\int_{\Omega}(u(t,x)+v(t,x))^{-q}dx)$ for some $q>0$.

In the rest of the introduction, we introduce some standing notations in
subsection 1.1, and state the main results of the paper and provide some
remarks on the main results in subsection 1.2.

###  1.1 Notations

In this subsection, we introduce some standing notations to be used throughout
the paper.

Observe that for given $u_{0}(\cdot)\in C^{0}(\bar{\Omega})$ and
$v_{0}(\cdot)\in C^{0}(\bar{\Omega})$ satisfying that $u_{0}\geq 0$,
$v_{0}\geq 0$, and $\int_{\Omega}(u_{0}(x)+v_{0}(x))dx>0$, if $v_{0}\equiv 0$
(resp. $u_{0}\equiv 0$), then $v(t,x;u_{0},v_{0})\equiv 0$ (resp.
$u(t,x;u_{0},v_{0})\equiv 0$) for $t\in(0,T_{\max}(u_{0},v_{0}))$ and
$(u(t,x;u_{0}),w(t,x;u_{0})):=(u(t,x;u_{0},0),w(t,x;u_{0},0))$ (resp.
$v(t,x;v_{0})$, $w(t,x;v_{0})):=(v(t,x;0,v_{0})$, $w(t,x;0,v_{0}))$) is the
solution of (1.3) (resp. (1.4)) with initial condition $u(0,x;u_{0})=u_{0}(x)$
(resp. $v(0,x;v_{0})=v_{0}(x)$). Hence, throughout the rest of this paper, we
consider classical solutions of (1.2) with the initial function
$u_{0}(x),v_{0}(x)$ satisfying

$u_{0},v_{0}\in C^{0}(\bar{\Omega}),\quad u_{0},v_{0}\geq 0,\quad{\rm
and}\quad\int_{\Omega}u_{0}>0,\,\,\,\int_{\Omega}v_{0}>0.$ (1.11)

Let

$\begin{cases}a_{\min}=\min\\{a_{1},a_{2}\\},\quad&a_{\max}=\max\\{a_{1},a_{2}\\}\cr
b_{\min}=\min\\{b_{1},b_{2}\\},\quad&b_{\max}=\max\\{b_{1},b_{2}\\}\cr
c_{\min}=\min\\{c_{1},c_{2}\\},\quad&c_{\max}=\max\\{c_{1},c_{2}\\}.\end{cases}$
(1.12)

For given $B>0$ and $\beta\not=|\chi_{2}-B|$, let

$f(\mu,\chi_{1},\chi_{2},\beta,B)=\mu(B+\beta)\Big{(}1+\frac{(|\chi_{2}-B|-\beta)^{2}+(\chi_{1}-\chi_{2})^{2}\beta}{4B\beta}\Big{)}.$
(1.13)

For fixed $\mu>0$, $\chi_{1}>0$ and $\chi_{2}>0$, let

$\displaystyle\chi_{1}^{*}(\mu,\chi_{1},\chi_{2})=\inf\Big{\\{}f(\mu,\chi_{1},\chi_{2},\beta,B)\,|\,$
$\displaystyle B>0,\beta>0,\beta\not=|\chi_{2}-B|\Big{\\}}.$ (1.14)

Similarly, let

$\displaystyle\chi_{2}^{*}(\mu,\chi_{1},\chi_{2})=\inf\Big{\\{}f(\mu,\chi_{2},\chi_{1},\beta,B)\,|\,$
$\displaystyle B>0,\beta>0,\beta\not=|\chi_{1}-B|\Big{\\}}.$ (1.15)

Let

$\chi^{*}(\mu,\chi_{1},\chi_{2})=\min\\{\chi_{1}^{*}(\mu,\chi_{1},\chi_{2}),\chi_{2}^{*}(\mu,\chi_{1},\chi_{2})\\}.$
(1.16)

The number $\chi^{*}(\mu,\chi_{1},\chi_{2})$ will be a lower bound for
$a_{\min}$ for the global boundedness of positive solutions of (1.2). We refer
the reader to Lemma 2.1 on the continuity and lower bounds of
$\chi^{*}(\mu,\chi_{1},\chi_{2})$.

###  1.2 Main results and remarks

In this subsection, we state the main results of the paper and provide some
remarks on the main results.

Observe that, for any given $u_{0},v_{0}$ satisfying (1.11), by the third
equation in (1.2),

$\mu\int_{\Omega}w(t,x;u_{0},v_{0})dx=\int_{\Omega}(\nu
u(t,x;u_{0},v_{0})+\lambda v(t,x;u_{0},v_{0}))dx$

and

$\displaystyle\min\\{\nu,\lambda\\}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx$
$\displaystyle\leq\int_{\Omega}(\nu u(t,x;u_{0},v_{0})+\lambda
v(t,x;u_{0},v_{0}))dx$
$\displaystyle\leq\max\\{\nu,\lambda\\}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx$

for any $t\in(0,T_{\max}(u_{0},v_{0}))$. Hence $\liminf_{t\nearrow
T_{\max}(u_{0},v_{0})}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx=0$
implies that (1.10) holds. On the other hand, we have

$w(t,x;u_{0},v_{0})\geq\delta_{0}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))>0\quad\forall\,x\in\Omega,\,\,t\in(0,T_{\max}(u_{0},v_{0}))$
(1.17)

for some positive constant $\delta_{0}$ independent of $u_{0},v_{0}$ (see
Lemma 2.4). Hence

$\liminf_{t\nearrow
T_{\max}(u_{0},v_{0})}\inf_{x\in\Omega}w(t,x;u_{0},v_{0})=0\,\,{\rm
iff}\,\,\liminf_{t\nearrow
T_{\max}(u_{0},v_{0})}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx=0.$

The first main theorem of the current paper is on the lower bounds of the
combined mass $\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx$ on any
bounded subinterval of $(0,T_{\max}(u_{0},v_{0}))$, which would provides the
upper bounds of
$\big{(}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx\big{)}^{-1}$ on
bounded subintervals of $(0,T_{\max}(u_{0},v_{0}))$.

######  Theorem 1.1 (Local lower bound of the combined mass).

For any $T\in(0,\infty)$ and any $u_{0},v_{0}$ satisfying (1.11),

$\inf_{0\leq
t<\min\\{T,T_{\max}(u_{0},v_{0})\\}}\int_{\Omega}\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}dx>0.$
(1.18)

######  Remark 1.1.

  * (1)

By Proposition 1.1, (1.17) and (1.18), if $T_{\max}(u_{0},v_{0})<\infty$, then
we must have

$\limsup_{t\nearrow
T_{\max}(u_{0},v_{0})}\left(\left\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\right\|_{C^{0}(\bar{\Omega})}\right)=\infty.$

  * (2)

If

$\inf_{0\leq
t<\min\\{T,T_{\max}(u_{0},v_{0})\\}}\int_{\Omega}u(t,x;u_{0},v_{0})dx>0$
(1.19)

and

$\inf_{0\leq
t<\min\\{T,T_{\max}(u_{0},v_{0})\\}}\int_{\Omega}v(t,x;u_{0},v_{0})dx>0,$
(1.20)

then (1.18) holds. But the converse may not be true because competitive
exclusion may occur in (1.2), which will be studied somewhere else.

The second main theorem of the current paper is on the local $L^{p}$\- and
$C^{\theta}$-boundedness of positive classical solutions of (1.2).

######  Theorem 1.2 (Local $L^{p}$\- and $C^{\theta}$-boundedness).

  * (1)

(Local $L^{p}$-boundedness) There are $\varepsilon_{0}>0$, $p>3N$ and
$M_{1}>0$ such that for any $T\in(0,\infty)$ and any $u_{0},v_{0}$ satisfying
(1.11),

$\displaystyle\int_{\Omega}(u+v)^{p}(t,x;u_{0},v_{0})dx\leq$ $\displaystyle
e^{-(t-\tau)}\int_{\Omega}(u+v)^{p}(\tau,x;u_{0},v_{0})dx$
$\displaystyle+M_{1}\Big{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x;u_{0},v_{0})}+1\Big{)}^{2(p+1)/\varepsilon_{0}}+M_{1}$

and

$\displaystyle\int_{\Omega}(u+v)^{p}(t,x;u_{0},v_{0})dx\leq\max\Big{\\{}$
$\displaystyle e^{-(t-\tau)}\int_{\Omega}(u+v)^{p}(\tau,x;u_{0},v_{0})dx,$
$\displaystyle M_{1}\Big{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x;u_{0},v_{0})}+1\Big{)}^{2(p+1)/\varepsilon_{0}}+M_{1}\Big{\\}}$

for any $0\leq\tau\leq t<\hat{T}$, where
$\hat{T}=\min\\{T,T_{\max}(u_{0},v_{0})\\}$.

  * (2)

(Local $C^{\theta}$-boundedness) For any $p>2N$ and $0<\theta<1-\frac{2N}{p}$,
there are $M_{2}>0$, $\beta>0$, and $\gamma>0$ such that for any $u_{0},v_{0}$
satisfying (1.11) and $T\in(0,\infty)$,

$\displaystyle\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{C^{\theta}(\bar{\Omega})}$
$\displaystyle\leq
M_{2}\Big{[}(t-\tau)^{-\beta}e^{-\gamma(t-\tau)}\|u(\tau,\cdot;u_{0},v_{0})+v(\tau,\cdot;u_{0},v_{0})\|_{L^{p}}$
$\displaystyle\,\,\,\,+\frac{\displaystyle\sup_{\tau\leq
t<\hat{T}}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{L^{p}}^{2}}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w(t,x;u_{0},v_{0})}$
$\displaystyle\,\,\,\,+\sup_{\tau\leq
t<\hat{T}}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{L^{p}}+1\Big{]}$
(1.21)

for any $0\leq\tau<\hat{T}=\min\\{T,T_{\max}(u_{0},v_{0})\\}$ and
$\tau<t<\hat{T}$.

######  Remark 1.2.

  * (1)

By Theorem 1.2, if $\inf_{x\in\Omega}w(t,x;u_{0},v_{0})$ is bounded away from
zero on $(0,T_{\max}(u_{0},v_{0}))$, then
$\int_{\Omega}(u^{p}(t,x;u_{0},v_{0})+v^{p}(t,x;u_{0},v_{0}))dx$ is bounded on
$(0,T_{\max}(u_{0},v_{0}))$ for some $p>3N$ and then
$\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{C^{\theta}(\bar{\Omega})}$
is bounded on $[\tau,T_{\max}(u_{0},v_{0}))$ for some $0<\theta<1$ and any
$0<\tau<T_{\max}(u_{0},v_{0})$.

  * (2)

By Theorems 1.1 and 1.2,
$\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{C^{\theta}(\bar{\Omega})}$
is bounded on $[\tau,\hat{T})$ for some $\theta\in(0,1)$, any
$T\in(0,\infty)$, and any $0<\tau<\hat{T}:=\min\\{T,T_{\max}(u_{0},v_{0})\\}$,
which plays an important role in the study of global existence of classical
solutions of (1.2).

The third main theorem of the current paper is on global existence of
classical solutions of (1.2).

######  Theorem 1.3 (Global existence).

For any $u_{0}$ and $v_{0}$ satisfying (1.11),
$(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0})$, $w(t,x;u_{0},v_{0}))$ exists
globally, that is, $T_{\max}(u_{0},v_{0})=\infty$.

######  Remark 1.3.

  * (1)

By (1.17), (1.18), and Theorem 1.1, for any $u_{0},v_{0}$ satisfying (1.11),

$\inf_{0\leq t<T}\inf_{x\in\Omega}w(t,x;u_{0},v_{0})\geq\delta_{0}\inf_{0\leq
t<T}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx>0\quad\forall\,0<T<\infty.$
(1.22)

  * (2)

By (1.22) and Theorem 1.2,

$\sup_{t\in(0,T)}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{L^{\infty}(\Omega)}<\infty\quad\forall\,0<T<\infty.$
(1.23)

  * (3)

It is interesting to see whether (1.22) and (1.23) also hold for $T=\infty$,
which is strongly related to the boundedness of
$\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))^{-q}$ for some $q>0$ and
the boundedness of $\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))^{p}$
for some $p>3N$, and is discussed in Theorems 1.4 and 1.5 in the following.

The fourth main theorem of the current paper is on the global boundedness of
$\int_{\Omega}(u+v)^{-q}$ and $\int_{\Omega}(u+v)^{p}$ for some $q>0$ and
$p>3N$. For given $u_{0},v_{0}$ satisfying (1.11) and $\tau\geq 0$, let

$m^{*}(\tau,u_{0},v_{0})={\rm
max}\Big{\\{}\int_{\Omega}u(\tau,x;u_{0},v_{0})dx,\frac{a_{1}|\Omega|}{b_{1}}\Big{\\}}+{\rm
max}\Big{\\{}\int_{\Omega}v(\tau,x;u_{0},v_{0})dx,\frac{a_{2}|\Omega|}{b_{2}}\Big{\\}}.$
(1.24)

######  Theorem 1.4.

Assume $a_{\min}>\chi^{*}(\mu,\chi_{1},\chi_{2})$. Then the following hold.

  * (1)

(Boundedness of $\int_{\Omega}(u+v)^{-q}$) There are $q>0$ and $M_{3}>0$ such
that for any $u_{0},v_{0}$ satisfying (1.11) and for every $0<\tau<\infty$,

$\displaystyle\int_{\Omega}(u(x,t;u_{0},v_{0})+v(x,t;u_{0},v_{0}))^{-q}dx$
$\displaystyle\leq
e^{-\frac{\epsilon_{0}q}{2}(t-\tau)}\int_{\Omega}(u(x,\tau;u_{0},v_{0})+v(x,\tau;u_{0},v_{0}))^{-q}dx+2qC_{\tau,u_{0},v_{0}}\epsilon_{0}^{-1}+M_{3}$

for all $t>\tau$, where
$\epsilon_{0}=a_{\min}-\chi^{*}(\mu,\chi_{1},\chi_{2})$ and

$\displaystyle C_{\tau,u_{0},v_{0}}=\begin{cases}0&\text{if}\;q\geq
1,\cr(b_{\max}+c_{\max})|\Omega|^{q}(m^{*}(\tau,u_{0},v_{0}))^{1-q}\;&\text{if}\;q<1.\end{cases}$

  * (2)

(Boundedness of $\int_{\Omega}(u+v)^{p}$) Let $p>3N$ and $\varepsilon_{0}>0$
be as in Theorem 1.2. Then there is $M_{4}>0$ such that for every $\tau>0$,

$\displaystyle\int_{\Omega}(u+v)^{p}(t,x;u_{0},v_{0})dx\leq$ $\displaystyle
e^{-(t-\tau)}\int_{\Omega}(u+v)^{p}(\tau,x;u_{0},v_{0})dx$
$\displaystyle+M_{4}\Bigg{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\infty,x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x;u_{0},v_{0})}+1\Bigg{)}^{2(p+1)/\varepsilon_{0}}+M_{4}$

for all $t>\tau$.

######  Remark 1.4.

The concrete estimates for
$\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))^{-q}dx$ and
$\int_{\Omega}(u(t,x;u_{0},v_{0})$ $+v(t,x;u_{0},v_{0}))^{p}dx$ obtained in
Theorem 1.4 provide useful tools for the study of the asymptotic behavior of
globally defined positive solutions of (1.2).

The last main theorem of the current paper is on uniform boundedness and
uniform pointwise persistence of classical solutions of (1.2), which provides
some insight into the understanding of the asymptotic behavior of globally
defined positive solutions of (1.2).

######  Theorem 1.5.

Assume $a_{\min}>\chi^{*}(\mu,\chi_{1},\chi_{2})$. Then the following hold.

  * (1)

(Uniform boundedness) Let $p>3N$ be as in Theorem 1.2 and $q>0$ be as in
Theorem 1.4(1). Let $0<\theta<\frac{N}{2p}$. There are $M_{1}^{*}>0$,
$M_{2}^{*}>0$ and $M_{3}^{*}>0$ such that for any $u_{0},v_{0}$ satisfying
(1.11),

$\limsup_{t\to\infty}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))^{-q}dx\leq
M_{1}^{*},$ (1.25)
$\limsup_{t\to\infty}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))^{p}dx\leq
M_{2}^{*},$ (1.26)

and

$\limsup_{t\to\infty}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{C^{\theta}(\bar{\Omega})}\leq
M_{3}^{*}.$ (1.27)

  * (2)

(Combined pointwise persistence) There is $M_{0}^{*}>0$ such that for any
$u_{0},v_{0}$ satisfying (1.11),

$\liminf_{t\to\infty}\inf_{x\in\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))\geq
M_{0}^{*}.$ (1.28)

######  Remark 1.5.

  * (1)

Let

$\displaystyle\mathcal{E}=\Big{\\{}u,v\in C^{0}(\bar{\Omega})\,|\,$
$\displaystyle u\geq 0,\,\,v\geq 0,\,\,\,\,\inf_{x\in\Omega}(u(x)+v(x))\geq
M_{0}^{*},\quad\text{and}$ $\displaystyle\int_{\Omega}(u(x)+v(x))^{-q}\leq
M_{1}^{*},\,\,\int_{\Omega}(u(x)+v(x))^{p}dx\leq M_{2}^{*}\Big{\\}}.$

Theorem 1.4 shows that $\mathcal{E}$ eventually attracts every globally
defined positive solution of (1.2).

  * (2)

(1.28) implies that if $a_{\min}>\chi^{*}(\mu,\chi_{1},\chi_{2})$, then there
is $m_{0}^{*}>0$ such that for any $u_{0},v_{0}$ satisfying (1.11),

$\liminf_{t\to\infty}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx\geq
m_{0}^{*},$ (1.29)

which is refereed to as combined mass persistence. It remains open whether
(1.29) holds without the assumption
$a_{\min}>\chi^{*}(\mu,\chi_{1},\chi_{2}).$

  * (3)

(1.27) implies that if $a_{\min}>\chi^{*}(\mu,\chi_{1},\chi_{2})$, then for
any $u_{0},v_{0}$ satisfying (1.11),

$\limsup_{t\to\infty}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{L^{\infty}(\Omega)}\leq
M_{3}^{*}.$ (1.30)

It also remains open whether (1.30) holds without the assumption
$a_{\min}>\chi^{*}(\mu,\chi_{1},\chi_{2}).$

  * (4)

We say both species persistent in mass if for any $u_{0},v_{0}$ satisfying
(1.11),

$\liminf_{t\to\infty}\int_{\Omega}u(t,x;u_{0},v_{0})dx>0\,\,{\rm
and}\,\,\liminf_{t\to\infty}\int_{\Omega}v(t,x;u_{0},v_{0})dx>0,$

and say competitive exclusion occurs in mass if for any $u_{0},v_{0}$
satisfying (1.11),

$\limsup_{t\to\infty}\int_{\Omega}u(t,x;u_{0},v_{0})dx=0\,\,{\rm
and}\,\,\liminf_{t\to\infty}\int_{\Omega}v(t,x;u_{0},v_{0})dx>0$

or

$\liminf_{t\to\infty}\int_{\Omega}u(t,x;u_{0},v_{0})dx>0\,\,{\rm
and}\,\,\limsup_{t\to\infty}\int_{\Omega}v(t,x;u_{0},v_{0})dx=0.$

We say both species persistent pointwiese if for any $u_{0},v_{0}$ satisfying
(1.11),

$\liminf_{t\to\infty}\inf_{x\in\Omega}u(t,x;u_{0},v_{0})>0\,\,{\rm
and}\,\,\liminf_{t\to\infty}\inf_{x\in\Omega}v(t,x;u_{0},v_{0})>0,$

and say competitive exclusion occurs pointwise if for any $u_{0},v_{0}$
satisfying (1.11),

$\limsup_{t\to\infty}\sup_{x\in\Omega}u(t,x;u_{0},v_{0})=0\,\,{\rm
and}\,\,\liminf_{t\to\infty}\inf_{x\in\Omega}v(t,x;u_{0},v_{0})>0$

or

$\liminf_{t\to\infty}\inf_{x\in\Omega}u(t,x;u_{0},v_{0})>0\,\,{\rm
and}\,\,\limsup_{t\to\infty}\sup_{x\in\Omega}v(t,x;u_{0},v_{0})=0.$

We will study the persistence of both species and competitive exclusion
somewhere else.

The rest of the paper is organized as follows. In section 2, we present some
preliminary lemmas. In section 3, we study local lower bound of
$\int_{\Omega}(u(t,x)+v(t,x))dx$; local upper bounds of
$\int_{\Omega}(u(t,x)+v(t,x))^{p}dx$ and
$\|u(t,\cdot)+v(t,\cdot)\|_{C^{\theta}(\bar{\Omega})}$ for some $p>3N$ and
$\theta\in(0,1)$; the global existence of classical solutions of (1.2), and
prove Theorems 1.1-1.3. We investigate the global boundedness of
$\int_{\Omega}(u+v)^{-q}$ and $\int_{\Omega}(u+v)^{p}$ and prove Theorems 1.4
and 1.5 in section 4. We prove two important technical propositions in the
Appendix.

##  2 Preliminary lemmas

In this section, we present some lemmas to be used in later sections.

First, let $\chi^{*}(\mu,\chi_{1},\chi_{2})$ be defined as in (1.16). We
present the following lemma on the continuity and lower bounds of
$\chi^{*}(\mu,\chi_{1},\chi_{2})$.

######  Lemma 2.1.

  * (1)

$\chi^{*}(\mu,\chi_{1},\chi_{2})$ is upper semicontinuous in $\mu>0$,
$\chi_{1}>0$ and $\chi_{2}>0$, that is,

$\limsup_{(\mu,\chi_{1},\chi_{2})\to(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})}\chi^{*}(\mu,\chi_{1},\chi_{2})\leq\chi^{*}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})\quad\forall\,\mu^{0}>0,\,\,\chi_{1}^{0}>0,\,\,\chi_{2}^{0}>0.$

  * (2)

For any $\mu>0$, $\chi_{1}>0$ and $\chi_{2}>0$,

$\chi^{*}(\mu,\chi_{1},\chi_{2})\leq\min\Big{\\{}\mu\chi_{2}+\frac{\mu(\chi_{1}-\chi_{2})^{2}}{4\chi_{2}},\mu\chi_{1}+\frac{\mu(\chi_{2}-\chi_{1})^{2}}{4\chi_{1}}\Big{\\}},$
(2.1)

and when $\chi_{1}=\chi_{2}=\chi$,

$\chi^{*}(\mu,\chi,\chi)=\begin{cases}\frac{\mu\chi^{2}}{4}&{\rm
if}\,\,0<\chi<2\cr\mu(\chi-1)&{\rm if}\,\,\chi\geq 2.\end{cases}$ (2.2)

###### Proof.

(1) We prove that $\chi^{*}(\mu,\chi_{1},\chi_{2})$ is upper semicontinuous in
$\mu>0$, $\chi_{1}>0$ and $\chi_{2}>0$.

Fix $(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})$ with for all
$\mu^{0},\chi_{1}^{0},\chi_{2}^{0}>0.$ Without loss of generality, we may
assume that
$\chi^{*}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})=\chi^{*}_{1}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0}).$
Note that for any $\epsilon>0,$ there are $\tilde{\beta}>0$ and $\tilde{B}>0$
satisfying $\tilde{\beta}\not=|\chi_{2}^{0}-\tilde{B}|$ such that

$\chi^{*}_{1}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})\geq
f(\mu^{0},\chi_{1}^{0},\chi_{2}^{0},\tilde{\beta},\tilde{B})-\epsilon$

and there is $\delta>0$ such that

$\tilde{\beta}\not=|\chi_{2}-\tilde{B}|$

and

$f(\mu,\chi_{1},\chi_{2},\tilde{\beta},\tilde{B})\leq
f(\mu^{0},\chi_{1}^{0},\chi_{2}^{0},\tilde{\beta},\tilde{B})+\epsilon$

for all $\mu>0,\chi_{1}>0,\chi_{2}>0$ satisfying $|\mu-\mu^{0}|<\delta$,
$|\chi_{1}-\chi_{1}^{0}|<\delta$, and $|\chi_{2}-\chi_{2}^{0}|<\delta$. This
implies that

$f(\mu,\chi_{1},\chi_{2},\tilde{\beta},\tilde{B})\leq
f(\mu^{0},\chi_{1}^{0},\chi_{2}^{0},\tilde{\beta},\tilde{B})+\epsilon\leq\chi^{*}_{1}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})+2\epsilon,$
(2.3)

and then

$\chi_{1}^{*}(\mu,\chi_{1},\chi_{2})\leq\chi^{*}_{1}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})+2\epsilon.$

Therefore,

$\limsup_{(\mu,\chi_{1},\chi_{2})\to(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})}\chi^{*}(\mu,\chi_{1},\chi_{2})\leq\chi^{*}(\mu^{0},\chi_{1}^{0},\chi_{2}^{0})\quad\forall\,\mu^{0},\chi_{1}^{0},\chi_{2}^{0}>0.$

(2) We first prove (2.2) when $\chi_{1}=\chi_{2}$. In this case, we have

$f(\mu,\chi_{1},\chi_{2},\beta,B)=\mu(B+\beta)\big{(}1+\frac{(|\chi_{2}-B|-\beta)^{2}}{4\beta}\big{)}.$

This implies that

$\displaystyle\chi_{1}^{*}(\mu,\chi_{1},\chi_{2})$
$\displaystyle=\inf\Big{\\{}\mu(B+\beta)\big{(}1+\frac{(|\chi_{2}-B|-\beta)^{2}}{4\beta}\big{)}\,|\,B>0,\,\beta>0,\beta\not=|\chi_{2}-B|\Big{\\}}$
$\displaystyle=\inf\Big{\\{}\mu(B+\beta)\big{(}1+\frac{(|\chi_{2}-B|-\beta)^{2}}{4\beta}\big{)}\,|\,B>0,\,0<\beta\leq|\chi_{2}-B|\Big{\\}}$
$\displaystyle=\inf\Big{\\{}\mu(B+\beta)\big{(}1+\frac{(\chi_{2}-B-\beta)^{2}}{4\beta}\big{)}\,|\,0<B<\chi_{2},\,0<\beta\leq\chi_{2}-B\Big{\\}}$
$\displaystyle=\inf\Big{\\{}\mu\beta\Big{(}1+\frac{(\chi_{2}-\beta)^{2}}{4\beta}\Big{)}\,|\,0<\beta<\chi_{2}\Big{\\}}$
$\displaystyle=\begin{cases}\frac{\mu\chi_{2}^{2}}{4}\quad&{\rm
if}\,\,0<\chi_{2}<2\cr\mu(\chi_{2}-1)\quad&{\rm if}\,\,\chi_{2}>2.\end{cases}$

Similarly,

$\chi_{2}^{*}(\mu,\chi_{1},\chi_{2})=\begin{cases}\frac{\mu\chi_{1}^{2}}{4}\quad&{\rm
if}\,\,0<\chi_{1}<2\cr\mu(\chi_{1}-1)\quad&{\rm if}\,\,\chi_{1}>2.\end{cases}$

Hence (2.2) holds.

Next we prove (2.1) for any $\mu>0$, $\chi_{1}>0$ and $\chi_{2}>0$. Note that

$f(\mu,\chi_{1},\chi_{2},\beta,B)=\frac{\mu}{4}\Big{(}\frac{1}{\beta}+\frac{1}{B}\Big{)}\Big{(}4B\beta+B(|\chi_{2}-B|-\beta)^{2}+(\chi_{1}-\chi_{2})^{2}\beta\Big{)}.$

Then

$\displaystyle\chi_{1}^{*}(\mu,\chi_{1},\chi_{2})\leq\inf\Big{\\{}f(\mu,\chi_{1},\chi_{2},\beta,\chi_{2})\,|\,\beta>0\Big{\\}}=\mu\chi_{2}+\frac{\mu(\chi_{1}-\chi_{2})^{2}}{4\chi_{2}}.$
(2.4)

Similarly,

$\displaystyle\chi_{2}^{*}(\mu,\chi_{1},\chi_{2})\leq\inf\Big{\\{}f(\mu,\chi_{2},\chi_{1},\beta,\chi_{1})\,|\,\beta>0\Big{\\}}=\mu\chi_{1}+\frac{\mu(\chi_{2}-\chi_{1})^{2}}{4\chi_{1}}.$
(2.5)

Hence (2.1) holds. ∎

Next, 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

$A_{p}=-\Delta+\mu I:D(A_{p})\subset L^{p}(\Omega)\to L^{p}(\Omega)$ (2.6)

with

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

Then $-A_{p}$ generates an analytic semigroup on $L^{p}(\Omega)$. We denote it
by $e^{-tA_{p}}$. Note that ${\rm Re}\sigma(A_{p})>0$. Let $A_{p}^{\beta}$ be
the fractional power operator of $A_{p}$ (see [11, Definition 1.4.1]). Let
$X_{p}^{\beta}=\mathcal{D}(A_{p}^{\beta})$ with graph norm
$\|u\|_{X_{p}^{\beta}}=\|A_{p}^{\beta}u\|_{L^{p}(\Omega)}$ for $\beta\geq 0$
and $u\in X_{p}^{\beta}$ (see [11, Definition 1.4.7]).

######  Lemma 2.2.

  * (i)

For each $p\in(1,\infty)$ and $\beta\geq 0$, there is $C_{p,\beta}>0$ such
that

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

for some $\gamma>0$.

  * (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 [11, Theorem 1.4.3].

(ii) It follows from [11, Theorem 1.6.1].

(iii) It also follows from [11, Theorem 1.6.1]. ∎

######  Lemma 2.3.

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

$\|A_{p}^{\beta}e^{-tA_{p}}\nabla\cdot w\|_{L^{p}(\Omega)}\leq
C_{p,\beta,\epsilon}t^{-\beta-\frac{1}{2}-\epsilon}e^{-\gamma
t}\|w\|_{L^{p}(\Omega)}\quad\text{for all}\,\,t>0\,\text{and some
}\,\gamma>0.$ (2.7)

Consequently, 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 [15, Lemma 2.1]. ∎

We then present some lemmas on the lower and upper bounds of the solutions of

$\begin{cases}-\Delta w+\mu w=\nu u+\lambda
v,\quad&x\in\Omega\cr\frac{\partial w}{\partial
n}=0,\quad&x\in\partial\Omega,\end{cases}$ (2.8)

where $\mu,\nu$, and $\lambda$ are positive constants. For given $u,v\in
L^{p}(\Omega)$, let $w(\cdot;u,v)$ be the solution of (2.8).

######  Lemma 2.4.

Let $u,v\in C^{0}(\bar{\Omega})$ be nonnegative function such that
$\int_{\Omega}u>0$ and $\int_{\Omega}v>0$. Then

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

where $\delta_{0}$ is some positive constant independent of $u,v$.

###### Proof.

It follows from the arguments of [10, Lemma 2.1] and the Gaussian lower bound
for the heat kernel of the laplacian with Neumann boundary condition on smooth
domain. ∎

######  Lemma 2.5.

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

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

###### Proof.

It follows from $L^{p}$-estimates for elliptic equations (see [1, Theorem
12.1]). ∎

######  Lemma 2.6.

For any nonnegative $u,v\in C(\bar{\Omega})$,

$\int_{\Omega}\frac{|\nabla
w(\cdot;u,v)|^{2}}{w(\cdot;u,v)^{2}}\leq\mu|\Omega|.$

###### Proof.

Multiplying (2.8) by $\frac{1}{w}$ and integrating it over $\Omega$ yields
that

$\displaystyle 0=\int_{\Omega}\frac{1}{w}\cdot\Big{(}\Delta w-\mu w+\nu
u+\lambda v\Big{)}=\int_{\Omega}\frac{|\nabla
w|^{2}}{w^{2}}-\mu|\Omega|+\nu\int_{\Omega}\frac{u}{w}+\lambda\int_{\Omega}\frac{v}{w}$

for all $t\in(0,T_{\max})$. The lemma thus follows. ∎

Throughout the rest of this section, we assume that $u_{0}(x)$ and $v_{0}(x)$
satisfies (1.11) and
$(u(t,x),v(t,x),w(t,x)):=(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0}))$
is the unique classical solution of (1.2) on the maximal interval
$(0,T_{\max}):=(0,T_{\max}(u_{0},v_{0}))$ with the initial condition
$(u(0,x),v(0,x))=(u_{0}(x),v_{0}(x))$. Note that

$u(t,x),v(t,x),w(t,x)>0\quad\forall\,x\in\Omega,\,\,t\in(0,T_{\max}).$

We now present some upper bound for $\int_{\Omega}u(t,x;u_{0},v_{0})dx$ and
$\int_{\Omega}v(t,x;u_{0},v_{0})dx$.

######  Lemma 2.7.

For any $\tau\in[0,T_{\max})$, there exist positive constants
$m^{*}_{1}(\tau,u_{0},v_{0}),m^{*}_{2}(\tau,u_{0},v_{0})>0$ such that

$\int_{\Omega}u(t,x;u_{0},v_{0})dx\leq m^{*}_{1}(\tau,u_{0},v_{0}):={\rm
max}\Big{\\{}\int_{\Omega}u(\tau,x;u_{0},v_{0})dx,\frac{a_{1}|\Omega|}{b_{1}}\Big{\\}}\quad\forall\,\,t\in[\tau,T_{\max}),$
(2.9)

and

$\int_{\Omega}v(t,x;u_{0},v_{0})dx\leq m^{*}_{2}(\tau,u_{0},v_{0})={\rm
max}\Big{\\{}\int_{\Omega}v(\tau,x;u_{0},v_{0})dx,\frac{a_{2}|\Omega|}{b_{2}}\Big{\\}}\quad\forall\,\,t\in[\tau,T_{\max}),$
(2.10)

for any $t\in[\tau,T_{\max})$, where $|\Omega|$ is the Lebesgue measure of
$\Omega$. Moreover, if $T_{\max}(u_{0},v_{0})=\infty$, then

$\limsup_{t\to\infty}\int_{\Omega}u(t,x;u_{0},v_{0})dx\leq\frac{a_{1}|\Omega|}{b_{1}}\quad{\rm
and}\quad\limsup_{t\to\infty}\int_{\Omega}v(t,x;u_{0},v_{0})dx\leq\frac{a_{2}|\Omega|}{b_{2}}.$
(2.11)

###### Proof.

By integrating the first equation in (1.2) with respect to $x$, we get that

$\displaystyle\frac{d}{dt}\int_{\Omega}u$ $\displaystyle=\int_{\Omega}\Delta
u-\chi\int_{\Omega}\nabla\cdot\Big{(}\frac{u}{w}\nabla
w\Big{)}+a_{1}\int_{\Omega}u-b_{1}\int_{\Omega}u^{2}-c_{1}\int_{\Omega}vu$
$\displaystyle\leq
a_{1}\int_{\Omega}u-\frac{b_{1}}{|\Omega|}\Big{(}\int_{\Omega}u\Big{)}^{2}\quad\forall\,t\in(0,T_{\max}).$

This together with comparison principle for scalar ODEs implies (2.9) and the
first inequality in (2.11) when $T_{\max}=\infty$.

Similarly, we can prove (2.10) and the second inequality in (2.11) when
$T_{\max}=\infty$. ∎

##  3 Proofs of Theorems 1.1-1.3

In this section, we investigate local lower bound of the combined mass and
local $L^{p}$-and $C^{\theta}$-boundedness of classical solutions of (1.2);
the global existence of classical solutions of (1.2); and prove Theorems
1.1-1.3.

Throughout this section, for given $u_{0},v_{0}$ satisfying (1.11), we put

$(u(t,x),v(t,x),w(t,x)):=(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0})),$

and if no confusion occurs, we may drop $(t,x)$ in $u(t,x)$ (resp. $v(t,x)$,
$w(t,x)$).

###  3.1 Lower bound of the combined mass on bounded intervals and proof of
Theorem 1.1

In this subsection, we study the lower bound of the combined mass on bounded
intervals and proof of Theorem 1.1.

We first present a lemma on $\int_{\Omega}\ln(u+v)dx$.

######  Lemma 3.1.

For any $u_{0}(x)$ and $v_{0}(x)$ satisfying (1.11), there exists
$K=K(u_{0},v_{0})>0$ such that

$\frac{d}{dt}\int_{\Omega}\ln\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}dx\geq-K\quad\text{\rm
for all}\,\,t\in(0,T_{\max}(u_{0},v_{0})).$

###### Proof.

Note that

$\displaystyle\frac{\partial}{\partial t}(u+v)=$
$\displaystyle\Delta(u+v)-\nabla\cdot\left(\frac{\chi_{1}u+\chi_{2}v}{w}\nabla
w\right)$
$\displaystyle+(a_{1}u+a_{2}v)-(b_{1}u^{2}+b_{2}v^{2})-(c_{1}+c_{2})uv.$ (3.1)

Multiplying the above equation by $\frac{1}{u+v}$ and then integrating on
$\Omega$ yields that

$\displaystyle\frac{d}{dt}\int_{\Omega}\ln{(u+v)}\geq$
$\displaystyle\int_{\Omega}\frac{|\nabla(u+v)|^{2}}{(u+v)^{2}}-\underbrace{\int_{\Omega}\frac{\chi_{1}u+\chi_{2}v}{(u+v)^{2}}\nabla(u+v)\cdot\frac{\nabla
w}{w}}_{I}$
$\displaystyle+\int_{\Omega}\frac{a_{1}u+a_{2}v}{u+v}-\int_{\Omega}\frac{b_{1}u^{2}+b_{2}v^{2}}{u+v}-\int_{\Omega}\frac{(c_{1}+c_{2})uv}{u+v}$

for all $t\in(0,T_{\max}(u_{0},v_{0})).$ By Young’s inequality and Lemma 2.6,
we have

$\displaystyle
I=\int_{\Omega}\frac{\chi_{1}u+\chi_{2}v}{(u+v)^{2}}\nabla(u+v)\cdot\frac{\nabla
w}{w}$
$\displaystyle\leq\int_{\Omega}\frac{|\nabla(u+v)|^{2}}{(u+v)^{2}}+\frac{\max\\{\chi_{1}^{2},\chi_{2}^{2}\\}}{4}\int_{\Omega}\frac{|\nabla
w|^{2}}{w^{2}}$
$\displaystyle\leq\int_{\Omega}\frac{|\nabla(u+v)|^{2}}{(u+v)^{2}}+\frac{\mu|\Omega|\max\\{\chi_{1}^{2},\chi_{2}^{2}\\}}{4}.$

Then by Lemma 2.7 we have

$\frac{d}{dt}\int_{\Omega}\ln{(u+v)}\geq-\frac{\mu|\Omega|\max\\{\chi_{1}^{2},\chi_{2}^{2}\\}}{4}+a_{\min}|\Omega|-\left(b_{\max}+c_{\max}\right)(m_{1}^{*}(0,u_{0},v_{0})+m_{2}^{*}(0,u_{0},v_{0}))$

for all $t\in(0,T_{\max})$. The lemma is thus proved. ∎

We now prove Theorem 1.1.

###### Proof of Theorem 1.1.

Fix a $\tau\in(0,T_{\max}(u_{0},v_{0}))$. It is clear that

$\inf_{0\leq
t\leq\tau}\int_{\Omega}\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))dx>0.$

It then suffices to prove that there exist $C=C(T)>0$ such that

$\int_{\Omega}\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}dx\geq
C(T)\quad\text{\rm for all}\,\,t\in(\tau,\hat{T}),$ (3.2)

where $\hat{T}=\min\\{T,T_{\max}(u_{0},v_{0})\\}$.

Note that
$L:=\int_{\Omega}\ln{\big{(}u(\tau,x;u_{0},v_{0})+v(\tau,x;u_{0},v_{0})\big{)}}dx$
is finite. By Lemma 3.1, there exists $K=K(u_{0},v_{0})>0$ such that

$\frac{d}{dt}\int_{\Omega}\ln{\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}dx}\geq-K$

for all $t\in(0,T_{\max}(u_{0},v_{0}))$. We thus have that

$\displaystyle\int_{\Omega}\ln{\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}}dx$
$\displaystyle\geq\int_{\Omega}\ln{(u(\tau,x;u_{0},v_{0})+v(\tau,x;u_{0},v_{0}))}dx-K\cdot(t-\tau)$
$\displaystyle\geq L-K\cdot(\hat{T}-\tau)=C(K,L,\tau)\quad\text{for
all}\,\,t\in(\tau,\hat{T}).$

Therefore Jensen’s inequality asserts that

$\displaystyle\int_{\Omega}\ln{\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}}dx$
$\displaystyle=|\Omega|\cdot\int_{\Omega}\ln{\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}}\frac{dx}{|\Omega|}$
$\displaystyle\leq|\Omega|\cdot\ln{\Big{(}\int_{\Omega}\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}\frac{dx}{|\Omega|}\Big{)}}$

for all $t\in(0,T_{\max}(u_{0},v_{0}))$, which implies

$\displaystyle\int_{\Omega}\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}dx$
$\displaystyle\geq|\Omega|\cdot\mathop{\hbox{exp}}{\Big{(}\frac{1}{|\Omega|}\cdot\int_{\Omega}\ln{\big{(}u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0})\big{)}}dx\Big{)}}$
$\displaystyle\geq|\Omega|\cdot e^{\frac{1}{|\Omega|}\cdot
C(K,L,\tau)}\quad\text{for all}\,\,t\in(\tau,\hat{T}),$

which implies (3.2). The theorem is thus proved. ∎

###  3.2 Local $L^{p}$\- and $C^{\theta}$-Boundedness and proof of Theorem 1.2

In this subsection, we study the $L^{p}$\- and $C^{\theta}$-boundedness of
classical solutions of (1.2) and prove Theorem 1.2. We first make some
observations.

Fix $p>2$. First, observe that

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+v)^{p}=$
$\displaystyle-(p-1)\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}$
$\displaystyle+(p-1)\int_{\Omega}(u+v)^{p-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle+\int_{\Omega}(u+v)^{p-1}(a_{1}u+a_{2}v)-\int_{\Omega}(u+v)^{p-1}(b_{1}u^{2}+b_{2}v^{2})-\int_{\Omega}(u+v)^{p-1}(c_{1}+c_{2})uv$
$\displaystyle\leq$
$\displaystyle-(p-1)\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}$
$\displaystyle+(p-1)\int_{\Omega}(u+v)^{p-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle+a_{\max}\int_{\Omega}(u+v)^{p}-\min\\{b_{\min},c_{\min}\\}\int_{\Omega}(u+v)^{p+1}$
(3.3)

for $t\in(0,T_{\max}(u_{0},v_{0}))$. To prove Theorem 1.1, it is then
essential to provide proper estimates for the integral
$\int_{\Omega}(u+v)^{p-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$.

Next, observe that

$\displaystyle\int_{\Omega}\frac{(u+v)^{p-2}(\chi_{1}u+\chi_{2}v)}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle=\chi_{2}\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\quad+(\chi_{1}-\chi_{2})\int_{\Omega}\frac{(u+v)^{p-2}}{w}u\nabla(u+v)\cdot\nabla
w.$

Since $u^{2}\leq(u+v)^{2},$ by Young’s inequality, for any $\varepsilon>0,$ we
have that

$\displaystyle\big{|}(\chi_{1}-\chi_{2})\int_{\Omega}\frac{(u+v)^{p-2}}{w}u\nabla(u+v)\cdot\nabla
w\big{|}$
$\displaystyle\leq\varepsilon\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}+\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\int_{\Omega}(u+v)^{p-2}u^{2}\frac{|\nabla
w|^{2}}{w^{2}}$
$\displaystyle\leq\varepsilon\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}+\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\int_{\Omega}(u+v)^{p}\frac{|\nabla
w|^{2}}{w^{2}}.$

Multiplying the third equation in (1.2) by $\frac{(u+v)^{p}}{w}$ and then
integrating over $\Omega$ yields that

$\displaystyle 0$
$\displaystyle=\int_{\Omega}\frac{(u+v)^{p}}{w}\cdot\big{(}\Delta w-\mu w+\nu
u+\lambda v\big{)}$
$\displaystyle=-\int_{\Omega}\frac{p(u+v)^{p-1}w\nabla(u+v)-(u+v)^{p}\nabla
w}{w^{2}}\cdot\nabla w$
$\displaystyle\quad-\mu\int_{\Omega}(u+v)^{p}+\nu\int_{\Omega}\frac{u(u+v)^{p}}{w}+\lambda\int_{\Omega}\frac{v(u+v)^{p}}{w}$

for all $t\in(0,T_{\max}(u_{0},v_{0})).$ Thus we have,

$\int_{\Omega}(u+v)^{p}\frac{|\nabla w|^{2}}{w^{2}}\leq
p\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla
w+\mu\int_{\Omega}(u+v)^{p}$

for all $t\in(0,T_{\max}(u_{0},v_{0})).$ Therefore, for any ${\varepsilon}>0$,

$\displaystyle\int_{\Omega}\frac{(u+v)^{p-2}(\chi_{1}u+\chi_{2}v)}{w}\nabla(u+v)\cdot\nabla
w$ $\displaystyle\leq\varepsilon\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}dx$
$\displaystyle\quad+\Big{(}\chi_{2}+\frac{p(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\Big{)}\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\quad+\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\mu\int_{\Omega}(u+v)^{p}.$
(3.4)

To provide proper estimates for
$\int_{\Omega}(u+v)^{p-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$, it is then essential to provide proper estimates for
$\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla w$. We have the
following two propositions on the estimates for
$\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla w$.

######  Proposition 3.1.

There is $p>3N$ such that for every given $\epsilon>0$ and
$0<\varepsilon_{0}\ll 1$, there is $C>0$ such that

$\displaystyle\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla w$
$\displaystyle\leq\epsilon\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}$
$\displaystyle\quad+C\Big{(}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+\int_{\Omega}\Big{(}\frac{|\nabla
w|}{w}\Big{)}^{2p+2-\varepsilon_{0}}\Big{)}+C$ (3.5)

for any $u_{0},v_{0}\in C^{0}(\bar{\Omega})$ satisfying (1.11), and
$t\in(0,T_{\max}(u_{0},v_{0}))$.

######  Proposition 3.2.

Let $p\geq 3$ and $p-\sqrt{2p-3}<k<p+\sqrt{2p-3}.$ There are $M,\tilde{M}>0$
such that

$\int_{\Omega}\frac{|\nabla w|^{2p}}{w^{k}}\leq M\int_{\Omega}\frac{(\nu
u+\lambda
v)^{p}}{w^{k-p}}+\tilde{M}\int_{\Omega}w^{2p-k}\quad\forall\,\,t\in(0,T_{\rm
max}(u_{0},v_{0})).$

In the rest of this subsection, we prove Theorem 1.2 by applying Propositions
3.1 and 3.2. Proposition 3.1 can be proved by the similar arguments as those
in [22, Proposition 4.1], and Proposition 3.2 can be proved by the similar
arguments as those in [22, Proposition 4.2]. For the reader’s convenience, we
will provide the outline of the proofs of Propositions 3.1 and 3.2 in the
appendix.

###### Proof of Theorem 1.2.

(1) First, let $p$ be as in Proposition 3.1. By (3.2) and (3.2), for any
$\varepsilon>0$, we have

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+v)^{p}$
$\displaystyle\leq-(p-1)\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}+(p-1)\varepsilon\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}dx$
$\displaystyle\quad+(p-1)\Big{(}\chi_{2}+\frac{p(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\Big{)}\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\quad+(p-1)\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\mu\int_{\Omega}(u+v)^{p}$
$\displaystyle\quad+a_{\max}\int_{\Omega}(u+v)^{p}-\min\\{b_{\min},c_{\min}\\}\int_{\Omega}(u+v)^{p+1}$
(3.6)

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

Next, let $\hat{T}=\min\\{T,T_{\max}(u_{0},v_{0})\\}$ and
$\tau\in(0,T_{\max}(u_{0},v_{0}))$. By (3.2) and Proposition 3.1, there are
$0<\varepsilon_{0}\ll 1$ and $C_{1}>0$ such that

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+v)^{p}$
$\displaystyle\leq
C_{1}\Big{(}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+\int_{\Omega}\Big{(}\frac{|\nabla
w|}{w}\Big{)}^{2p+2-\varepsilon_{0}}\Big{)}+C_{1}$
$\displaystyle\quad+(p-1)\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\mu\int_{\Omega}(u+v)^{p}$
$\displaystyle\quad+a_{\max}\int_{\Omega}(u+v)^{p}-\min\\{b_{\min},c_{\min}\\}\int_{\Omega}(u+v)^{p+1}$
$\displaystyle\leq\frac{C_{1}}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}\int_{\Omega}\frac{|\nabla
w|^{2p+2-\varepsilon_{0}}}{w^{p+1-\varepsilon_{0}/2}}$
$\displaystyle\quad+C_{1}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+C_{1}+(p-1)\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\mu\int_{\Omega}(u+v)^{p}$
$\displaystyle\quad+a_{\max}\int_{\Omega}(u+v)^{p}-\min\\{b_{\min},c_{\min}\\}\int_{\Omega}(u+v)^{p+1}$
(3.7)

for all $t\in[\tau,\hat{T})$. This together with Proposition 3.2 implies that
there is $C_{2}>0$ such that

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+v)^{p}$
$\displaystyle\leq\frac{C_{2}}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}\Big{(}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}/2}+\int_{\Omega}w^{p+1-\varepsilon_{0}/2}\Big{)}$
$\displaystyle\quad+C_{1}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+C_{1}+(p-1)\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\mu\int_{\Omega}(u+v)^{p}$
$\displaystyle\quad+a_{\max}\int_{\Omega}(u+v)^{p}-\min\\{b_{\min},c_{\min}\\}\int_{\Omega}(u+v)^{p+1}$
(3.8)

for all $t\in[\tau,\hat{T})$.

Now, by Lemma 2.5 and the Young’s inequality, there is
$C_{3}\geq\max\\{C_{1},C_{2}\\}$ such that

$\
\int_{\Omega}w^{p+1-\varepsilon_{0}/2}\leq\frac{C_{3}}{C_{2}}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}/2}\,\,{\rm
and}\,\,\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}\leq\frac{C_{3}}{C_{1}}\int_{\Omega}(u+v)^{p+1-\epsilon_{0}/2}+C_{3}.$

This together with (3.2) implies that

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+v)^{p}$
$\displaystyle\leq\Bigg{(}\frac{2C_{3}}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}+C_{3}\Bigg{)}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}/2}$
$\displaystyle\quad+C_{1}+C_{3}+(p-1)\frac{(\chi_{1}-\chi_{2})^{2}}{4\varepsilon}\mu\int_{\Omega}(u+v)^{p}$
$\displaystyle\quad+a_{\max}\int_{\Omega}(u+v)^{p}-\min\\{b_{\min},c_{\min}\\}\int_{\Omega}(u+v)^{p+1}$
(3.9)

for $t\in[\tau,\hat{T})$. This together with the Young’s inequality implies
that there is a positive constant $C_{4}>0$ such that

$\displaystyle\frac{d}{dt}\int_{\Omega}(u+v)^{p}(t,x)dx\leq$
$\displaystyle-\int_{\Omega}(u+v)^{p}(t,x)dx$
$\displaystyle+C_{4}\Bigg{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}+1\Bigg{)}^{2(p+1)/\varepsilon_{0}}+C_{4}$
(3.10)

for all $t\in[\tau,\hat{T})$.

Finally, by (3.2) and the comparison principle for scalar ODEs, we have

$\displaystyle\int_{\Omega}(u+v)^{p}(t,x)dx\leq$ $\displaystyle
e^{-(t-\tau)}\int_{\Omega}(u+v)^{p}(\tau,x)dx$
$\displaystyle+C_{4}\bigg{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}+1\Bigg{)}^{2(p+1)/\varepsilon_{0}}+C_{4}$

and

$\displaystyle\int_{\Omega}(u+v)^{p}(t,x)dx$
$\displaystyle\leq\max\left\\{e^{-(t-\tau)}\int_{\Omega}(u+v)^{p}(\tau,x)dx,C_{4}\Bigg{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\hat{T},x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}+1\Bigg{)}^{2(p+1)/\varepsilon_{0}}+C_{4}\right\\}$

for any $t\in[\tau,\hat{T})$. Theorem 1.2(1) thus follows with $M_{1}:=C_{4}$.

(2) For given $p>2N$ and $0<\theta<1-\frac{2N}{p}$, choose
$\beta\in(0,\frac{1}{2})$ such that $2\beta-\frac{2N}{p}>\theta.$ By Lemma
2.2, there is $C_{p,\beta}>0$ such that for any $u_{0},v_{0}$ satisfying
(1.11),

$\|u(\cdot,\cdot)+v(\cdot,\cdot)\|_{C^{\theta}(\bar{\Omega})}\leq
C_{p,\beta}\|u(t,\cdot)+v(t,\cdot)\|_{X_{\frac{p}{2}}^{\beta}}\quad\forall\,0<t<T_{\max}.$
(3.11)

Note that in view of (3.1), we have that

$\displaystyle\|u(t,\cdot)+v(t,\cdot)\|_{X_{\frac{p}{2}}^{\beta}}$
$\displaystyle\leq\|A_{\frac{p}{2}}^{\beta}e^{-A_{\frac{p}{2}}(t-\tau)}(u(\tau,\cdot;u_{0},v_{0})+v(\tau,\cdot;u_{0},v_{0}))\|_{L^{p/2}(\Omega)}$
$\displaystyle\quad+\int_{\tau}^{t}\left\|A_{\frac{p}{2}}^{\beta}e^{-A_{\frac{p}{2}}(t-s)}\nabla\cdot\left(\frac{\chi_{1}u(s,\cdot)+\chi_{2}v(s,\cdot)}{w(s,\cdot)}\nabla
w(s,\cdot)\right)\right\|_{L^{p/2}(\Omega)}ds$
$\displaystyle\quad+\int_{\tau}^{t}\|A_{\frac{p}{2}}^{\beta}e^{-A_{\frac{p}{2}}(t-s)}g(u(s,\cdot),v(s,\cdot))\|_{L^{p/2}(\Omega)}ds$
(3.12)

for any $0\leq\tau<T_{\max}(u_{0},v_{0})$ and $\tau<t<T_{\max}(u_{0},v_{0})$,
where $A_{\frac{p}{2}}=-\Delta+\mu I$ is defined as in (2.6) with $p$ being
replaced by $\frac{p}{2}$, and

$g(u,v)=(\mu+a_{1})u+(\mu+a_{2})v-(b_{1}u^{2}+b_{2}v^{2})-(c_{1}+c_{2})uv.$

By Lemma 2.2 again, there is $\gamma>0$ such that

$\displaystyle\|A_{\frac{p}{2}}^{\beta}e^{-A_{\frac{p}{2}}(t-\tau)}(u(\tau,\cdot;u_{0},v_{0})+v(\tau,\cdot;u_{0},v_{0}))\|_{L^{p/2}(\Omega)}$
$\displaystyle\leq
C_{p,\beta}(t-\tau)^{-\beta}e^{-\gamma(t-\tau)}\|u(\tau,\cdot;u_{0},v_{0})+v(\tau,\cdot;u_{0},v_{0})\|_{L^{p/2}}\quad\forall\,{t>\tau}.$
(3.13)

By Lemma 2.3 and Lemma 2.5, for any $\epsilon>0$, there are $\gamma>0$,
$C_{p}>0$ and $C_{p,\beta,\epsilon}>0$ such that

$\displaystyle\int_{\tau}^{t}\left\|A_{\frac{p}{2}}^{\beta}e^{-A+{\frac{p}{2}}(t-s)}\nabla\cdot\left(\frac{\chi_{1}u(s,\cdot)+\chi_{2}v(s,\cdot)}{w(s,\cdot)}\nabla
w(s,\cdot)\right)\right\|_{L^{p/2}(\Omega)}ds$ $\displaystyle\leq
C_{p,\beta,\epsilon}\int_{\tau}^{t}(1+(t-s)^{-\beta-\frac{1}{2}-\epsilon})e^{-\gamma(t-s)}\left\|\frac{\chi_{1}u(s,\cdot)+\chi_{2}v(s,\cdot)}{w(s,\cdot)}\cdot\nabla
w(s,\cdot)\right\|_{L^{p/2}(\Omega)}ds$
$\displaystyle\leq{C_{p,\beta,\epsilon}(\chi_{1}+\chi_{2})}\int_{\tau}^{t}\frac{(1+(t-s)^{-\beta-\frac{1}{2}-\epsilon})e^{-\gamma(t-s)}\left\|u(s,\cdot)+v(s,\cdot)\right\|_{L^{p}(\Omega)}\left\|\nabla
w(s,\cdot)\right\|_{L^{p}(\Omega)}}{\displaystyle\inf_{\tau\leq
t<\min\\{T,T_{\max}\\},x\in\Omega}w(t,x)}ds$
$\displaystyle\leq{C_{p}C_{p,\beta,\epsilon}(\chi_{1}+\chi_{2})}\int_{\tau}^{t}\frac{(1+(t-s)^{-\beta-\frac{1}{2}-\epsilon})e^{-\gamma(t-s)}\left\|u(s,\cdot)+v(s,\cdot)\right\|^{2}_{L^{p}(\Omega)}}{\displaystyle\inf_{\tau\leq
t<\min\\{T,T_{\max}\\},x\in\Omega}w(t,x)}ds.$ (3.14)

By Lemma 2.2 again,

$\displaystyle\int_{\tau}^{t}\|A_{\frac{p}{2}}^{\beta}e^{-A_{\frac{p}{2}}(t-s)}g(u(s,\cdot),v(s,\cdot))\|_{L^{p/2}(\Omega)}ds$
$\displaystyle\leq
C_{p,\beta}\int_{\tau}^{t}(t-s)^{-\beta}e^{-\gamma(t-s)}\Big{\\{}\mu|\Omega|^{2/p}+a_{\max}\|u(s,\cdot)+v(s,\cdot)\|_{L^{p/2}}+b_{\max}\|(u(s,\cdot)+v(s,\cdot))^{2}\|_{L^{p/2}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{c_{\max}}{2}\|(u(s,\cdot)+v(s,\cdot))^{2}\|_{L^{p/2}}\Big{\\}}ds.$
(3.15)

((2)) then follows from (3.11) to (3.2). ∎

###  3.3 Global existence and proof of Theorem 1.3

In this subsection, we study the global existence of classical solutions of
(1.2) and prove Theorem 1.3.

###### Proof of Theorem 1.3.

We prove Theorem 1.3 by contradiction. Assume that $T_{\max}<\infty$. Then by
Lemma 2.4 and Theorem 1.1, there is $\delta>0$ such that

$w(t,x)\geq\delta\quad\quad\text{\rm for all}\,\,x\in\Omega\quad\text{\rm
and}\,\,t\in(0,T_{\max}),$ (3.16)

and then by Proposition 1.1, we have

$\limsup_{t\nearrow
T_{\max}}\left\|u(t,\cdot)+v(t,\cdot)\right\|_{C^{0}(\bar{\Omega})}=\infty.$

But, by (3.16) and Theorem 1.1, we have

$\lim_{t\nearrow
T_{\max}}\left\|u(t,\cdot)+v(t,\cdot)\right\|_{C^{0}(\bar{\Omega})}<\infty,$

which is a contradiction. Therefore $T_{\max}=\infty$. ∎

##  4 Proofs of Theorems 1.4 and 1.5

In this section, we study the boundedness of $\int_{\Omega}(u+v)^{-q}$ and
$\int_{\Omega}(u+v)^{p}$ and prove Theorems 1.4 and 1.5.

Throughout this section, for given $u_{0},v_{0}$ satisfying (1.11), we put

$(u(t,x),v(t,x),w(t,x)):=(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0})),$

and if no confusion occurs, we may drop $(t,x)$ in $u(t,x)$ (resp. $v(t,x)$,
$w(t,x)$).

###  4.1 Proof of Theorem 1.4

In this subsection, we study the boundedness of $\int_{\Omega}(u+v)^{-q}$ and
$\int_{\Omega}(u+v)^{p}$ and prove Theorem 1.4.

We first prove a lemma.

######  Lemma 4.1.

Let $\mu>0$, $\chi_{1}>0$ and $\chi_{2}>0$ be given.

  * (1)

For any $B>0$ and $0<\beta\not=|\chi_{2}-B|$,

$\displaystyle\frac{1}{q}\frac{d}{dt}\int_{\Omega}(u+v)^{-q}\leq$
$\displaystyle\Big{(}f(\mu,\chi_{1},\chi_{2},\beta,B)-a_{\min}\Big{)}\int_{\Omega}(u+v)^{-q}+b_{\max}\int_{\Omega}(u+v)^{-q+1}$
$\displaystyle+c_{\max}\int_{\Omega}(u+v)^{-q+1}\quad\forall\,{t\in(0,\infty)},$
(4.1)

where $f(\mu,\chi_{1},\chi_{2},\beta,B)$ is as in (1.13) and

$q=\frac{4B\beta}{B(|\chi_{2}-B|-\beta)^{2}+(\chi_{1}-\chi_{2})^{2}\beta}.$
(4.2)

  * (2)

For any $B>0$ and $0<\beta\not=|\chi_{1}-B|$,

$\displaystyle\frac{1}{q}\frac{d}{dt}\int_{\Omega}(u+v)^{-q}\leq$
$\displaystyle\Big{(}f(\mu,\chi_{2},\chi_{1},\beta,B)-a_{\min}\Big{)}\int_{\Omega}(u+v)^{-q}+b_{\max}\int_{\Omega}(u+v)^{-q+1}$
$\displaystyle+c_{\max}\int_{\Omega}(u+v)^{-q+1}\quad\forall\,{t\in(0,\infty)},$
(4.3)

where

$q=\frac{4B\beta}{B(|\chi_{1}-B|-\beta)^{2}+(\chi_{1}-\chi_{2})^{2}\beta}.$
(4.4)

###### Proof.

(1) First, multiplying (3.1) by $(u+v)^{-q-1}$ and integrating over $\Omega$,
we get

$\displaystyle\frac{1}{q}\frac{d}{dt}\int_{\Omega}(u+v)^{-q}=$
$\displaystyle-(q+1)\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle+(q+1)\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle-\int_{\Omega}(u+v)^{-q-1}(a_{1}u+a_{2}v)+\int_{\Omega}(u+v)^{-q-1}(b_{1}u^{2}+b_{2}v^{2})$
$\displaystyle+\int_{\Omega}(u+v)^{-q-1}(c_{1}+c_{2})uv$ $\displaystyle\leq$
$\displaystyle-(q+1)\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle+(q+1)\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$ $\displaystyle-
a_{\min}\int_{\Omega}(u+v)^{-q}+b_{\max}\int_{\Omega}(u+v)^{-q+1}+c_{\max}\int_{\Omega}(u+v)^{-q+1}.$
(4.5)

Next, we provide some estimate for
$\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$. Observe that

$\displaystyle\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle=\chi_{2}\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\quad+(\chi_{1}-\chi_{2})\int_{\Omega}(u+v)^{-q-2}\frac{u}{w}\nabla(u+v)\cdot\nabla
w$ (4.6)

for any $q>0$. Observe also that for any $B>0$ and $q>0$,

$\displaystyle(\chi_{1}-\chi_{2})\int_{\Omega}(u+v)^{-q-2}\frac{u}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\leq\frac{(\chi_{1}-\chi_{2})^{2}q}{4B}\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle\quad+\frac{B}{q}\int_{\Omega}(u+v)^{-q-2}u^{2}\frac{|\nabla
w|^{2}}{w^{2}}$
$\displaystyle\leq\frac{(\chi_{1}-\chi_{2})^{2}q}{4B}\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle\quad+\frac{B}{q}\int_{\Omega}(u+v)^{-q}\frac{|\nabla
w|^{2}}{w^{2}}.$ (4.7)

Multiplying the third equation in (1.2) by $\frac{(u+v)^{-q}}{w}$ and then
integrating over $\Omega$ with respect to $x$, we obtain that

$\displaystyle 0$
$\displaystyle=\int_{\Omega}\frac{(u+v)^{-q}}{w}\cdot\big{(}\Delta w-\mu w+\nu
u+\lambda v\big{)}$
$\displaystyle=-\int_{\Omega}\frac{(-q)(u+v)^{-q-1}w\nabla(u+v)-(u+v)^{-q}\nabla
w}{w^{2}}\cdot\nabla w$
$\displaystyle\quad-\mu\int_{\Omega}(u+v)^{-q}+\nu\int_{\Omega}\frac{u(u+v)^{-q}}{w}+\lambda\int_{\Omega}\frac{v(u+v)^{-q}}{w}$

for all ${t\in(0,\infty)}.$ Thus we have,

$q\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w+\int_{\Omega}(u+v)^{-q}\frac{|\nabla
w|^{2}}{w^{2}}\leq\mu\int_{\Omega}(u+v)^{-q}$ (4.8)

for all $t\in(0,\infty).$ It then follows that

$\frac{B}{q}\int_{\Omega}(u+v)^{-q}\frac{|\nabla
w|^{2}}{w^{2}}\leq\frac{B\mu}{q}\int_{\Omega}(u+v)^{-q}-B\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w.$

This together with (4.1) and (4.1) implies that

$\displaystyle\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\leq(\chi_{2}-B)\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\quad+\frac{(\chi_{1}-\chi_{2})^{2}q}{4B}\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle\quad+\frac{B\mu}{q}\int_{\Omega}(u+v)^{-q}$ (4.9)

for any $B>0$ and $q>0$.

Now, let $\tilde{\chi}=|\chi_{2}-B|$. Note that, when $B>0$ and $q>0$ are such
that $D:=\frac{(\chi_{1}-\chi_{2})^{2}}{4B}q<1$, for any $\beta>0$, by Young’s
inequality and (4.8),

$\displaystyle\tilde{\chi}\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle=(\tilde{\chi}-\beta)\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\quad+\beta\int_{\Omega}\frac{(u+v)^{-q-1}}{w}\nabla(u+v)\cdot\nabla
w$
$\displaystyle\leq(1-D)\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}+\frac{|\tilde{\chi}-\beta|^{2}}{4(1-D)}\int_{\Omega}(u+v)^{-q}\frac{|\nabla
w|^{2}}{w^{2}}$
$\displaystyle\quad+\frac{\beta\mu}{q}\int_{\Omega}(u+v)^{-q}-\frac{\beta}{q}\int_{\Omega}(u+v)^{-q}\frac{|\nabla
w|^{2}}{w^{2}}.$

This together with (4.1) implies that

$\displaystyle\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$ $\displaystyle\leq\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle\quad+\Big{(}\frac{B|\tilde{\chi}-\beta|^{2}}{4B-(\chi_{1}-\chi_{2})^{2}q}-\frac{\beta}{q}\Big{)}\int_{\Omega}(u+v)^{-q}\frac{|\nabla
w|^{2}}{w^{2}}$
$\displaystyle\quad+\Big{(}\frac{B\mu}{q}+\frac{\beta\mu}{q}\Big{)}\int_{\Omega}(u+v)^{-q}$
(4.10)

for any $\beta>0$ and $B>0$, $q>0$ with
$\frac{(\chi_{1}-\chi_{2})^{2}}{4B}{q}<1$. For any $B>0$ and $\beta>0$,
$\beta\not=\tilde{\chi}$, let $q$ be as in (4.2). Then

$\frac{(\chi_{1}-\chi_{2})^{2}}{4B}q<1,$ (4.11)

and

$\frac{B|\tilde{\chi}-\beta|^{2}}{4B-(\chi_{1}-\chi_{2})^{2}q}-\frac{\beta}{q}=0.$
(4.12)

Hence we have

$\displaystyle\int_{\Omega}(u+v)^{-q-2}\frac{\chi_{1}u+\chi_{2}v}{w}\nabla(u+v)\cdot\nabla
w$ $\displaystyle\leq\int_{\Omega}(u+v)^{-q-2}|\nabla(u+v)|^{2}$
$\displaystyle\quad+\Big{(}\frac{B\mu}{q}+\frac{\beta\mu}{q}\Big{)}\int_{\Omega}(u+v)^{-q}.$
(4.13)

Finally, assume that (4.11) and (4.12) hold. Then by (4.1) and (4.1), we have

$\displaystyle\frac{1}{q}\frac{d}{dt}\int_{\Omega}(u+v)^{-q}\leq$
$\displaystyle\Big{(}({B\mu+\beta\mu})\Big{(}1+\frac{1}{q}\Big{)}-a_{\min}\Big{)}\int_{\Omega}(u+v)^{-q}$
$\displaystyle+b_{\max}\int_{\Omega}(u+v)^{-q+1}+c_{\max}\int_{\Omega}(u+v)^{-q+1}.$
(4.14)

Note that

$(B\mu+\beta\mu)\Big{(}1+\frac{1}{q}\Big{)}=f(\mu,\chi_{1},\chi_{2},\beta,B).$

This implies that ((1)) holds.

(2) It can be proved by the similar arguments as in (1). ∎

We now prove Theorem 1.4.

###### Proof.

(1) First, without loss of generality, we assume that

$\chi^{*}(\mu,\chi_{1},\chi_{2})=\min\\{\chi_{1}^{*}(\mu,\chi_{1},\chi_{2}),\chi_{2}^{*}(\mu,\chi_{1},\chi_{2})\\}=\chi_{1}^{*}(\mu,\chi_{1},\chi_{2}).$

Let ${\epsilon_{0}}=a_{\min}-\chi^{*}(\mu,\chi_{1},\chi_{2})$. Then by the
definition of $\chi_{1}^{*}(\mu,\chi_{1},\chi_{2})$, there are $B>0$ and
$0<\beta\not=|\chi_{2}-B|$ such that

$a_{\min}-f(\mu,\chi_{1},\chi_{2},\beta,B)>\frac{3\epsilon_{0}}{4}.$

By ((1)), we then have

$\displaystyle\frac{1}{q}\frac{d}{dt}\int_{\Omega}(u+v)^{-q}\leq-\frac{3{\epsilon_{0}}}{4}\int_{\Omega}(u+v)^{-q}+b_{\max}\int_{\Omega}(u+v)^{-q+1}+c_{\max}\int_{\Omega}(u+v)^{-q+1}$
(4.15)

for all $t>0$, where $q$ is as in (4.2).

Next, note that if $q\geq 1$, by Young’s inequality, there is
$C_{\epsilon_{0}}>0$ such that

$\displaystyle b_{\max}\int_{\Omega}(u+v)^{-q+1}\leq$
$\displaystyle\frac{{\epsilon_{0}}}{8}\int_{\Omega}(u+v)^{-q}+C_{\epsilon_{0}},$
(4.16) $\displaystyle c_{\max}\int_{\Omega}(u+v)^{-q+1}\leq$
$\displaystyle\frac{\epsilon_{0}}{8}\int_{\Omega}(u+v)^{-q}+C_{\epsilon_{0}},$
(4.17)

and if $q<1$, then by Hölder inequality

$\displaystyle b_{\max}\int_{\Omega}(u+v)^{-q+1}\leq$ $\displaystyle
b_{\max}|\Omega|^{q}\cdot\left(\int_{\Omega}u+v\right)^{1-q}\leq
b_{\max}|\Omega|^{q}(m^{*}(\tau,u_{0},v_{0}))^{1-q},$ (4.18) $\displaystyle
c_{\max}\int_{\Omega}(u+v)^{-q+1}\leq$ $\displaystyle
c_{\max}|\Omega|^{q}\cdot\left(\int_{\Omega}u+v\right)^{1-q}\leq
c_{\max}|\Omega|^{q}(m^{*}(\tau,u_{0},v_{0})^{1-q}$ (4.19)

for any $t>\tau\geq 0$, where
$m^{*}(\tau,u_{0},v_{0})=m_{1}^{*}(\tau,u_{0},v_{0})+m_{2}^{*}(\tau,u_{0},v_{0})$.
By (4.15)-(4.18), there is $\tilde{C}_{\epsilon_{0},\tau,u_{0},v_{0}}>0$ such
that

$\frac{1}{q}\frac{d}{dt}\int_{\Omega}(u+v)^{-q}\leq-\frac{\epsilon_{0}}{2}\int_{\Omega}(u+v)^{-q}+\tilde{C}_{\epsilon_{0},\tau,u_{0},v_{0}}\quad\forall\,t>\tau>0,$

where $\tilde{C}_{\epsilon_{0},\tau,u_{0},v_{0}}=2C_{\epsilon_{0}}$ if $q\geq
1$ and
$\tilde{C}_{\epsilon_{0},\tau,u_{0},v_{0}}=(b_{\max}+c_{\max})|\Omega|^{q}(m^{*}(\tau,u_{0},v_{0}))^{1-q}$
if $q<1$. This implies that

$\displaystyle\int_{\Omega}(u(x,t;u_{0},v_{0})+v(x,t;u_{0},v_{0}))^{-q}dx$
$\displaystyle\leq
e^{-\frac{\epsilon_{0}q}{2}(t-\tau)}\int_{\Omega}(u(x,\tau;u_{0},v_{0})+v(x,\tau;u_{0},v_{0}))^{-q}dx+2q\tilde{C}_{\epsilon_{0},\tau,u_{0},v_{0}}\epsilon_{0}^{-1}$
(4.20)

for any $\tau>0$ and $t>\tau$. Theorem 1.4(1) then follows with

$\displaystyle M_{3}=\begin{cases}2C_{\epsilon_{0}}\;&{\rm if}\;q\geq 1\cr
0\;&{\rm if}\;q<1\end{cases}\;{\rm and}$ $\displaystyle
C_{\tau,u_{0},v_{0}}=\begin{cases}0\;&{\rm if}\;q\geq
1\cr(b_{\max}+c_{\max})|\Omega|^{q}(m^{*}(\tau,u_{0},v_{0}))^{1-q}\;&{\rm
if}\;q<1.\end{cases}$ (4.21)

(2) Let $M_{4}=M_{1}$. Theorem 1.4(2) follows from the arguments of Theorem
1.2. ∎

###  4.2 Proof of Theorem 1.5

In this subsection, we investigate the ultimate upper bounds of
$\int_{\Omega}(u+v)^{-q}$, $\int_{\Omega}(u+v)^{p}$, and $\|u+v\|_{\infty}$
for globally defined classical solutions of (1.2), ultimate lower bound of
$\inf_{x\in\Omega}(u+v)$, and prove Theorem 1.5.

###### Proof of Theorem 1.5.

(1) First, we prove (1.25). Let
$\epsilon_{0}=a_{\min}-\chi^{*}(\mu,\chi_{1},\chi_{2})$. By Theorem 1.4(1),
there is $M_{3}>0$ such that

$\displaystyle\int_{\Omega}(u(x,t;u_{0},v_{0})+v(x,t;u_{0},v_{0}))^{-q}dx$
$\displaystyle\leq
e^{-\frac{\epsilon_{0}q}{2}(t-\tau)}\int_{\Omega}(u(x,\tau;u_{0},v_{0})+v(x,\tau;u_{0},v_{0}))^{-q}dx+M_{3}+2qC_{\tau,u_{0},v_{0}}\epsilon_{0}^{-1}$
(4.22)

for all $t>\tau$, where $C_{\tau,u_{0},v_{0}}$ is as in (4.21). By Lemma 2.7,

$\limsup_{\tau\to\infty}\int_{\Omega}u(\tau,x;u_{0},v_{0})\leq\frac{a_{1}|\Omega|}{b_{1}}\quad{\rm
and}\quad\limsup_{\tau\to\infty}\int_{\Omega}v(\tau,x;u_{0},v_{0})dx\leq\frac{a_{2}|\Omega|}{b_{2}}.$

This implies that there is $\tau>0$ such that

$C_{\tau,u_{0},v_{0}}\leq
2(b_{\max}+c_{\max})|\Omega|^{q}\Big{(}\frac{a_{1}|\Omega|}{b_{1}}+\frac{a_{2}|\Omega|}{b_{2}}\Big{)}^{1-q}.$

This together with (4.2) implies that

$\limsup_{t\to\infty}\int_{\Omega}(u(t,x;u_{0},v_{0})+v(t,x;u_{0},v_{0}))^{-q}\leq
M_{1}^{*},$

where

$M_{1}^{*}=M_{3}+2(b_{\max}+c_{\max})|\Omega|^{q}\Big{(}\frac{a_{1}|\Omega|}{b_{1}}+\frac{a_{2}|\Omega|}{b_{2}}\Big{)}^{1-q}.$

(1.25) is thus proved.

Next, we prove (1.26). By Theorem 1.4(2), for every $\tau>0$,

$\displaystyle\int_{\Omega}(u+v)^{p}(t,x;u_{0},v_{0})dx\leq$ $\displaystyle
e^{-(t-\tau)}\int_{\Omega}(u+v)^{p}(\tau,x;u_{0},v_{0})dx$
$\displaystyle+M_{4}\Bigg{(}\frac{1}{\displaystyle\inf_{\tau\leq
t<\infty,x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x;u_{0},v_{0})}+1\Bigg{)}^{2(p+1)/\varepsilon_{0}}+M_{4}$
(4.23)

for all $t>\tau$. Note that

$\displaystyle|\Omega|=\int_{\Omega}(u+v)^{\frac{q}{q+1}}(u+v)^{-\frac{q}{q+1}}dx\leq\Big{(}\int_{\Omega}(u+v)\Big{)}^{\frac{q}{q+1}}\Big{(}\int_{\Omega}(u+v)^{-q}\Big{)}^{\frac{1}{q+1}}.$

This implies that

$\int_{\Omega}(u+v)\geq|\Omega|^{\frac{q+1}{q}}\Big{(}\int_{\Omega}(u+v)^{-q}\Big{)}^{-\frac{1}{q}}.$

Then by Lemma 2.4,

$w(t,x)\geq\delta_{0}\int_{\Omega}(u(t,x)+v(t,x))dx\geq\delta_{0}|\Omega|^{\frac{q+1}{q}}\Big{(}\int_{\Omega}(u+v)^{-q}(t,x)dx\Big{)}^{-\frac{1}{q}}$
(4.24)

for all $t>0$ and $x\in\Omega$. This implies that

$\displaystyle\frac{1}{\displaystyle\inf_{\tau\leq
t<\infty,x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}$
$\displaystyle=\sup_{\tau\leq
t<\infty,x\in\Omega}\frac{1}{w^{p+1-\varepsilon_{0}/2}(t,x)}$
$\displaystyle\leq\Bigg{(}\sup_{\tau\leq
t<\infty,x\in\Omega}\frac{(\int_{\Omega}(u+v)^{-q}(t,x)dx)^{\frac{1}{q}}}{\delta_{0}|\Omega|^{\frac{q+1}{q}}}\Bigg{)}^{p+1-\varepsilon_{0}/2}.$

By (1.25), there is $\tau>0$ such that

$\displaystyle\frac{1}{\displaystyle\inf_{\tau\leq
t<\infty,x\in\Omega}w^{p+1-\varepsilon_{0}/2}(t,x)}\leq\left(\frac{(2M_{1}^{*})^{\frac{1}{q}}}{\delta_{0}|\Omega|^{\frac{q+1}{q}}}\right)^{p+1-\varepsilon_{0}/2}.$

This together with (4.2) implies that

$\limsup_{t\to\infty}\int_{\Omega}(u(t,x)+v(t,x))^{p}\leq M_{2}^{*},$

where

$M_{2}^{*}=M_{4}\left(\left(\frac{(2M_{1}^{*})^{\frac{1}{q}}}{\delta_{0}|\Omega|^{\frac{q+1}{q}}}\right)^{p+1-\varepsilon_{0}/2}+1\right)^{2(p+1)/\varepsilon_{0}}+M_{4}.$

(1.26) is thus proved.

Now, we prove (1.27). By the arguments of Theorem 1.2(2), we have

$\displaystyle\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{C^{\theta}(\bar{\Omega})}$
$\displaystyle\leq
M_{2}\Big{[}(t-\tau)^{-\beta}e^{-\gamma(t-\tau)}\|u(\tau,\cdot;u_{0},v_{0})+v(\tau,\cdot;u_{0},v_{0})\|_{L^{p}}$
$\displaystyle\,\,\,\,+\frac{\displaystyle\sup_{\tau\leq
t<\hat{T}}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{L^{p}}^{2}}{\displaystyle\inf_{\tau\leq
t<\min\\{T,T_{\max}(u_{0},v_{0})\\},x\in\Omega}w(t,x;u_{0},v_{0})}$
$\displaystyle\,\,\,\,+\sup_{\tau\leq
t<\hat{T}}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{L^{p}}+1\Big{]}$
(4.25)

This together with (1.25), (1.26), and (4.24) implies that

$\limsup_{t\to\infty}\|u(t,\cdot;u_{0},v_{0})+v(t,\cdot;u_{0},v_{0})\|_{C^{\theta}(\bar{\Omega})}\leq
M_{3}^{*},$

where

$M_{3}^{*}=M_{2}\left(\frac{2M_{1}^{*}M_{2}^{*}}{\delta_{0}|\Omega|^{\frac{q+1}{q}}}+M_{2}^{*}+1\right).$

(1.27) is thus proved.

(2) We prove it by contradiction. Assume that there is no $M_{0}^{*}>0$ such
that (1.28) holds. Then for any $n\in\mathbb{N}$, there are $u_{n},v_{n}$
satisfying (1.11) such that

$\liminf_{t\to\infty}\inf_{x\in\Omega}(u(t,x;0,u_{n},v_{n})+v(t,x;0,u_{n},v_{n}))\leq
m_{n}:=\frac{1}{n}.$ (4.26)

By Theorem 1.4(1) and (4.24), there are $T_{n}>0$ and $\tilde{M}_{1}^{*}$ such
that

$\int_{\Omega}u(t,x;u_{n},v_{n})+v(t,x;u_{n},v_{n})dx\geq{\tilde{M}_{1}^{*}}\quad\forall\,t\geq
T_{n}$ (4.27)

and

$\|u(t,x;u_{n},v_{n})+v(t,x;u_{n},v_{n})\|_{C^{\theta}(\bar{\Omega})}\leq
2M^{*}_{3}\quad\forall\,t\geq T_{n}.$ (4.28)

By (4.26), there are $t_{n}\in\mathbb{R}$ with $t_{n}\geq T_{n}+1$ and
$x_{n}\in\bar{\Omega}$ such that

$u(t_{n},x_{n};u_{n},v_{n})+v(t_{n},x_{n};u_{n},v_{n})\leq\frac{2}{n}.$ (4.29)

Note that

$\displaystyle u(t_{n},x;u_{n},v_{n})$
$\displaystyle=u(1,x;u(t_{n}-1,\cdot;u_{n},v_{n}),v(t_{n}-1,u_{n},v_{n}))$
(4.30)

and

$\displaystyle v(t_{n},x;0,u_{n},v_{n})$
$\displaystyle=v(1,x;u(t_{n}-1,\cdot;u_{n},v_{n}),v(t_{n}-1,\cdot;u_{n},v_{n})).$
(4.31)

By (4.28), without loss of generality, we may assume that there are
$u_{0}^{*},v_{0}^{*},u_{1}^{*},v_{1}^{*}\in C(\bar{\Omega})$ such that

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

and

$\lim_{n\to\infty}u(t_{n},x;u_{n},v_{n})=u_{1}^{*}(x),\quad\lim_{n\to\infty}v(t_{n},x;u_{n},v_{n}))=v_{1}^{*}(x).$

uniformly in $x\in\bar{\Omega}.$ Then, by (4.30) and (4.31), we have that

$u_{1}^{*}(x)+v_{1}^{*}(x)=u(1,x;u_{0}^{*},v_{0}^{*})+v(1,x;u_{0}^{*},v_{0}^{*}).$
(4.32)

By (4.27), (4.28), and the Dominated Convergence Theorem,

$\int_{\Omega}u_{0}^{*}(x)+v_{0}^{*}(x)=\lim_{n\to\infty}\int_{\Omega}(u(t_{n}-1,x;0,u_{n},v_{n})+v(t_{n}-1,x;0,u_{n},v_{n}))dx\geq{\tilde{M}_{1}^{*}}.$

Then by (4.32) and comparison principles for parabolic equations,

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

This together with (4.29) implies that

$\frac{2}{n}\geq\inf_{x\in\Omega}(u(t_{n},x;0,u_{n},v_{n})+v(t_{n},x;0,u_{n},v_{n}))\geq\frac{1}{2}\inf_{x\in\Omega}(u_{1}^{*}(x)+v_{1}^{*}(x))\quad\forall\,n\gg
1,$

which is a contradiction. Therefore (2) holds. ∎

##  5 Appendix: Proofs of Propositions 3.1 and 3.2

In this section, we outline the proofs of Propositions 3.1 and 3.2.

Throughout this section, for given $u_{0},v_{0}$ satisfying (1.11), we put

$(u(t,x),v(t,x),w(t,x)):=(u(t,x;u_{0},v_{0}),v(t,x;u_{0},v_{0}),w(t,x;u_{0},v_{0})),$

and if no confusion occurs, we may drop $(t,x)$ in $u(t,x)$ (resp. $v(t,x)$,
$w(t,x)$).

###  5.1 Outline of the proof of Proposition 3.1

Proposition 3.1 can be proved by the similar arguments as those in [22,
Proposition 4.1]. For the reader’s convenience, we provide some outline of the
proof of Proposition 3.1 in the following.

###### Outline of the proof of Proposition 3.1.

First of all, observe that if $c,d,r,l>0$ are positive constants such that
$cd-c-d>0$ and $p-l-r-1>0$, then by Young’s inequality, we have

$\displaystyle\int_{\Omega}\frac{(u+v)^{p-1}}{w}\nabla(u+v)\cdot\nabla w$
$\displaystyle=\int_{\Omega}(u+v)^{l}\nabla(u+v)\cdot(u+v)^{r}\frac{\nabla
w}{w}\cdot(u+v)^{p-l-r-1}$
$\displaystyle\leq\frac{1}{c}\underbrace{\int_{\Omega}(u+v)^{lc}|\nabla(u+v)|^{c}}_{I_{1}}+\frac{1}{d}\underbrace{\int_{\Omega}(u+v)^{rd}\Big{(}\frac{|\nabla
w|}{w}\Big{)}^{d}}_{I_{2}}$ $\displaystyle\quad+\frac{cd-
c-d}{cd}\underbrace{\int_{\Omega}((u+v)^{p-l-r-1})^{\frac{cd}{cd-
c-d}}}_{I_{3}}.$ (5.1)

Next, by [22, Lemmas 4.1, 4.2], there are positive numbers $1<c<2$, $d,l,r>0$,
and $p>\max\\{3N,l+r+1\\}$ such that the following hold,

$cd-c-d>0,$ (5.2) $m:=\frac{(2l-p+2)c}{2}>0,$ (5.3) $\frac{2m}{2-c}<p+1,$
(5.4) $0<\frac{d(p+1)}{p+1-rd}<2p+2,$ (5.5)

and

$\frac{cd(p-l-r-1)}{cd-c-d}<p+1.$ (5.6)

For any $0<\varepsilon_{0}\ll 1$, we then have

$p+1-\varepsilon_{0}>\max\Big{\\{}\frac{2m}{2-c},rd,\frac{cd(p-l-r-1)}{cd-
c-d}\Big{\\}}$ (5.7)

and

$2p+2-\varepsilon_{0}>\frac{d(p+1-\varepsilon_{0})}{p+1-\varepsilon_{0}-rd}.$
(5.8)

Now, for the above $c,d,l,r,p$ and $m$, we have $m>0$ and $lc-m>0$. Then by
Young’s inequality, for any $C_{0}>0$,

$\displaystyle I_{1}$ $\displaystyle=\int_{\Omega}(u+v)^{lc}|\nabla(u+v)|^{c}$
$\displaystyle=\int_{\Omega}(u+v)^{m}\cdot(u+v)^{lc-m}|\nabla(u+v)|^{c}$
$\displaystyle\leq\frac{2-c}{2}\Big{(}\frac{1}{2C_{0}}\Big{)}^{\frac{c}{2-c}}\int_{\Omega}((u+v)^{m})^{\frac{2}{2-c}}+cC_{0}\int_{\Omega}\Big{(}(u+v)^{lc-m}|\nabla(u+v)|^{c}\Big{)}^{\frac{2}{c}}$
$\displaystyle=\frac{2-c}{2}\Big{(}\frac{1}{2C_{0}}\Big{)}^{\frac{c}{2-c}}\int_{\Omega}(u+v)^{\frac{2m}{2-c}}+cC_{0}\int_{\Omega}(u+v)^{2l-\frac{2m}{c}}|\nabla(u+v)|^{2}$

for $t\in(0,T_{\max}(u_{0},v_{0}))$. By the definition of $m$,
$2l-\frac{2m}{c}=p-2.$ By (5.7) and Young’s inequality again, we have

$\displaystyle I_{1}$
$\displaystyle\leq\frac{2-c}{2}\Big{(}\frac{1}{2C_{0}}\Big{)}^{\frac{c}{2-c}}\int_{\Omega}(u+v)^{\frac{2m}{2-c}}+{cC_{0}}\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}$
$\displaystyle\leq
C+C\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+cC_{0}\int_{\Omega}(u+v)^{p-2}|\nabla(u+v)|^{2}$

for $t\in(0,T_{\max}(u_{0},v_{0}))$ and some $C>0$ (independent of $u$ and
$v$).

Applying Young’s inequality, we have that

$\displaystyle I_{2}$
$\displaystyle=\int_{\Omega}(u+v)^{rd}\left(\frac{|\nabla w|}{w}\right)^{d}$
$\displaystyle\leq\frac{rd}{p+1-\varepsilon_{0}}\int_{\Omega}((u+v)^{rd})^{\frac{p+1-\varepsilon_{0}}{rd}}+\frac{p+1-\varepsilon_{0}-rd}{p+1-\varepsilon_{0}}\int_{\Omega}\left(\frac{|\nabla
w|}{w}\right)^{d\cdot{\frac{p+1-\varepsilon_{0}}{p+1-\varepsilon_{0}-rd}}}$
$\displaystyle=\frac{rd}{p+1-\varepsilon_{0}}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+\frac{p+1-\varepsilon_{0}-rd}{p+1-\varepsilon}\int_{\Omega}\left(\frac{|\nabla
w|}{w}\right)^{{\frac{d(p+1-\varepsilon_{0})}{p+1-\varepsilon_{0}-rd}}}$

for $t\in(0,T_{\max}(u_{0},v_{0}))$. Then by (5.8) and Young’s inequality
again,

$\displaystyle I_{2}\leq
C\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+C\int_{\Omega}\left(\frac{|\nabla
w|}{w}\right)^{2p+2-\varepsilon_{0}}+C$

for $t\in(0,T_{\max}(u_{0},v_{0}))$ and some $C>0$ (independent of $u$ and
$v$).

By (5.7) and Young’s inequality, we have

$\displaystyle I_{3}$
$\displaystyle=\int_{\Omega}((u+v)^{p-l-r-1})^{\frac{cd}{cd-c-d}}$
$\displaystyle=\int_{\Omega}(u+v)^{\frac{cd(p-l-r-1)}{cd-c-d}}$
$\displaystyle\leq\frac{cd(p-l-r-1)}{(p+1-\varepsilon_{0})(cd-
c-d)}\int_{\Omega}(u+v)^{p+1-\varepsilon_{0}}+C$

for $t\in(0,T_{\max}(u_{0},v_{0}))$ and some $C>0$ (independent of $u$ and
$v$). The proposition then follows from (5.1) and the above estimates for
$I_{1}$, $I_{2}$, and $I_{3}$. ∎

###  5.2 Outline of the proof of Proposition 3.2

Proposition 3.2 can be proved by the similar arguments as those in [22,
Proposition 4.2]. For the reader’s convenience, we provide a proof of
Proposition 3.2 in the following.

We first present some lemmas.

######  Lemma 5.1.

Let $p\geq 3$ and $k\geq 2.$ Then

$(k-1)\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}\leq(p-1)\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}+\mu\int_{\Omega}\frac{|\nabla w|^{2p-2}}{w^{k-2}}$
(5.9)

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

###### Proof.

Multiplying the third equation in (1.2) by $\frac{|\nabla w|^{2p-2}}{w^{k-1}}$
and integrating it over $\Omega$ yields that

$\displaystyle(k-1)\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}+\nu\int_{\Omega}\frac{u|\nabla
w|^{2p-2}}{w^{k-1}}+\lambda\int_{\Omega}\frac{v|\nabla w|^{2p-2}}{w^{k-1}}$
$\displaystyle=(p-1)\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}+\mu\int_{\Omega}\frac{|\nabla w|^{2p-2}}{w^{k-2}}$

for all $t\in(0,T_{\max}(u_{0},v_{0}))$. This implies (5.9). ∎

To prove Proposition 3.2, it is then essential to provide proper estimate for
the integral $\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}$. By Young’s inequality, for any
$0<\varepsilon<p-\frac{3}{2}$, we have

$\displaystyle\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}$
$\displaystyle\leq\frac{p-\frac{3}{2}-\varepsilon}{p+k-3}\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}$
$\displaystyle\quad+\frac{p+k-3}{4\left(p-\frac{3}{2}-\varepsilon\right)}\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}.$ (5.10)

By a direct calculation, we have

$\displaystyle(p-2)\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}$
$\displaystyle=\int_{\partial\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-2}}\frac{\partial|\nabla
w|^{2}}{\partial\nu}+(k-2)\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}$ $\displaystyle\quad-\int_{\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-2}}\Delta|\nabla w|^{2}.$ (5.11)

Note that

$\Delta|\nabla w|^{2}=2\nabla w\cdot\nabla(\Delta w)+{2|D^{2}w|^{2}}=2\nabla
w\cdot\nabla(\mu w-\nu u-\lambda v)+{2|D^{2}w|^{2}},$

and

$\big{|}\nabla|\nabla w|^{2}\big{|}^{2}=2\sum_{i=1}^{n}|\nabla w\cdot\nabla
w_{x_{i}}|^{2}\leq 4|\nabla w|^{2}\sum_{i=1}^{n}{|\nabla
w_{x_{i}}|^{2}}=4|\nabla w|^{2}|D^{2}w|^{2}.$

It then follows that

$\displaystyle-\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-2}}\Delta|\nabla
w|^{2}$ $\displaystyle\leq 2\int_{\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-2}}\nabla(\nu u+\lambda v)\cdot\nabla w$
$\displaystyle\quad-2\mu\int_{\Omega}\frac{|\nabla
w|^{2p-2}}{w^{k-2}}-\frac{1}{2}\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}.$ (5.12)

By (5.2) and (5.2), we have

$\displaystyle\Big{(}p-\frac{3}{2}\Big{)}\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}$
$\displaystyle\leq\int_{\partial\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-2}}\frac{\partial|\nabla
w|^{2}}{\partial\nu}+(k-2)\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}$ $\displaystyle\quad+2\int_{\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-2}}\nabla(\nu u+\lambda v)\cdot\nabla
w-2\mu\int_{\Omega}\frac{|\nabla w|^{2p-2}}{w^{k-2}}.$ (5.13)

To get proper estimate for $\int_{\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-1}}\nabla w\cdot\nabla|\nabla w|^{2}$, it is then essential to
provide proper estimates for the integrals $\int_{\partial\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-2}}\frac{\partial|\nabla w|^{2}}{\partial\nu}$ and
$\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-2}}\nabla(\nu u+\lambda
v)\cdot\nabla w$.

######  Lemma 5.2.

For every $\epsilon>0$, there is $C>0$ such that

$\int_{\partial\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-2}}\frac{\partial|\nabla
w|^{2}}{\partial\nu}\leq\epsilon\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla
w|^{2}\big{|}^{2}+C\int_{\Omega}\frac{|\nabla w|^{2p-2}}{w^{k-2}},$

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

###### Proof.

It follows from the similar arguments in [22, Lemma 4.4]. ∎

######  Lemma 5.3.

Let $p\geq 3$ and $k\geq p.$ For every given $\varepsilon>0,$ there is $M>0$
such that

$\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-2}}\nabla(\nu u+\lambda
v)\cdot\nabla w\leq M\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}+{\varepsilon}\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla
w|^{2}\big{|}^{2}+{\varepsilon}\int_{\Omega}\frac{|\nabla w|^{2p}}{w^{k}}$

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

###### Proof.

First, we have that

$\displaystyle\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-2}}\nabla(\nu
u+\lambda v)\cdot\nabla w$
$\displaystyle=\underbrace{(k-2)\int_{\Omega}\frac{(\nu u+\lambda v)|\nabla
w|^{2p-2}}{w^{k-1}}}_{I_{1}}-\underbrace{\int_{\Omega}\frac{(\nu u+\lambda
v)|\nabla w|^{2p-4}}{w^{k-2}}\Delta w}_{I_{2}}$
$\displaystyle\,\,-\underbrace{(p-2)\int_{\Omega}\frac{(\nu u+\lambda
v)|\nabla w|^{2p-6}}{w^{k-2}}\nabla w\cdot\nabla|\nabla w|^{2}}_{I_{3}}.$

Next, by Young’s inequality, for every $B_{1}>0$, there exists a positive
constant $A_{1}=A_{1}(k,p,B_{1})>0$ such that

$\displaystyle I_{1}=(k-2)\int_{\Omega}\frac{(\nu u+\lambda v)|\nabla
w|^{2p-2}}{w^{k-1}}$ $\displaystyle=(k-2)\int_{\Omega}\frac{\nu u+\lambda
v}{w^{\frac{k-p}{p}}}\cdot\frac{|\nabla w|^{2p-2}}{w^{\frac{k(p-1)}{p}}}$
$\displaystyle\leq A_{1}\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}+B_{1}\int_{\Omega}\frac{|\nabla w|^{2p}}{w^{k}}.$ (5.14)

Now, by the fact that $\Delta w=\mu w-\nu u-\lambda v$, and Young’s
inequality, for every $B_{2}>0$, there exists a positive constant
$A_{2}=A_{2}(k,p,\nu,\lambda,B_{2})>0$ such that

$\displaystyle{-I_{2}}$ $\displaystyle=-\int_{\Omega}\frac{(\nu u+\lambda
v)|\nabla w|^{2p-4}}{w^{k-2}}\Delta w$
$\displaystyle=-\mu\int_{\Omega}\frac{(\nu u+\lambda v)|\nabla
w|^{2p-4}}{w^{k-3}}+\int_{\Omega}\frac{(\nu u+\lambda v)^{2}|\nabla
w|^{2p-4}}{w^{k-2}}$ $\displaystyle\leq\int_{\Omega}\frac{(\nu u+\lambda
v)^{2}}{w^{\frac{2(k-p)}{p}}}\cdot\frac{|\nabla
w|^{2p-4}}{w^{\frac{k(p-2)}{p}}}$ $\displaystyle\leq
A_{2}\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}+B_{2}\int_{\Omega}\frac{|\nabla w|^{2p}}{w^{k}}.$ (5.15)

Finally, for every ${\varepsilon}>0$ and $B_{3}>0$, there are positive
constants $A_{3}=A_{3}({\varepsilon},k,p,\nu,B_{3})>0$ and
$A_{4}=A_{4}(A_{4},B_{3})$ such that

$\displaystyle{-I_{3}}$ $\displaystyle=-(p-2)\int_{\Omega}\frac{(\nu u+\lambda
v)|\nabla w|^{2p-6}}{w^{k-2}}\nabla w\cdot\nabla|\nabla w|^{2}$
$\displaystyle\leq A_{3}\int_{\Omega}\frac{(\nu u+\lambda v)^{2}|\nabla
w|^{2p-4}}{w^{k-2}}+{{\varepsilon}}\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}$ $\displaystyle\leq
A_{4}\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}+B_{3}\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}+{{\varepsilon}}\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}.$ (5.16)

Combining (5.2), (5.2) and (5.2) with
$B_{1}=B_{2}=B_{3}=\frac{1}{3}\varepsilon$ and
$M=M(\varepsilon,k,p,\nu):=A_{1}+A_{2}+A_{4}$ completes the proof. ∎

We now prove Proposition 3.2.

###### Proof of Proposition 3.2.

First, by (5.2) and Lemmas 5.2 and 5.3, for any $0<\varepsilon<p-\frac{3}{2}$,
there are $M,C>0$ such that

$\displaystyle\left(p-\frac{3}{2}\right)\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla w|^{2}\big{|}^{2}$
$\displaystyle\leq\varepsilon\int_{\Omega}\frac{|\nabla
w|^{2p-6}}{w^{k-2}}\big{|}\nabla|\nabla
w|^{2}\big{|}^{2}+\varepsilon\int_{\Omega}\frac{|\nabla w|^{2p}}{w^{k}}$
$\displaystyle\quad+(k-2)\int_{\Omega}\frac{|\nabla w|^{2p-4}}{w^{k-1}}\nabla
w\cdot\nabla|\nabla w|^{2}$ $\displaystyle\quad+M\int_{\Omega}\frac{(\nu
u+\lambda v)^{p}}{w^{k-p}}+C\int_{\Omega}\frac{|\nabla w|^{2p-2}}{w^{k-2}}.$
(5.17)

Next, by (5.2) and (5.2), we have

$\displaystyle\frac{p-1}{p+k-3}\int_{\Omega}\frac{|\nabla
w|^{2p-4}}{w^{k-1}}\nabla w\cdot\nabla|\nabla w|^{2}$
$\displaystyle\leq\frac{\varepsilon}{p+k-3}\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}+\frac{M}{p+k-3}\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}$ $\displaystyle\,\,+\frac{C}{p+k-3}\int_{\Omega}\frac{|\nabla
w|^{2p-2}}{w^{k-2}}+\frac{p+k-3}{4\left(p-\frac{3}{2}-\varepsilon\right)}\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}.$ (5.18)

Now, by (5.9), and (5.2), for any $0<\varepsilon<p-\frac{3}{2}$, there holds

$\displaystyle\Big{(}k-1-\frac{(p+k-3)^{2}}{4(p-\frac{3}{2}-\varepsilon)}-\varepsilon\Big{)}\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}$ $\displaystyle\leq M\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}+(C+\mu)\int_{\Omega}\frac{|\nabla w|^{2p-2}}{w^{k-2}}.$
(5.19)

By Young’s inequality, for any $0<\varepsilon<p-\frac{3}{2}$, there is
$\tilde{M}(\varepsilon,p)$ such that

$(C+\mu)\int_{\Omega}\frac{|\nabla
w|^{2p-2}}{w^{k-2}}\leq\varepsilon\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}+\tilde{M}\int_{\Omega}w^{2p-k}.$ (5.20)

By (5.2) and (5.20),

$\displaystyle\Big{(}k-1-\frac{(p+k-3)^{2}}{4(p-\frac{3}{2}-\varepsilon)}-2\varepsilon\Big{)}\int_{\Omega}\frac{|\nabla
w|^{2p}}{w^{k}}$ $\displaystyle\leq M\int_{\Omega}\frac{(\nu u+\lambda
v)^{p}}{w^{k-p}}+\tilde{M}\int_{\Omega}w^{2p-k}.$ (5.21)

Finally, since $p-\sqrt{2p-3}\leq k<p+\sqrt{2p-3}$, one can find
$\varepsilon>0$ such that

$k-1-2\varepsilon-\frac{(p+k-3)^{2}}{4\left(p-\frac{3}{2}-\varepsilon\right)}>0.$

The proposition then follows from (5.2). ∎

## References

  * [1] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 593-676.
  * [2] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ Math J. 65 (2016), 553-583.
  * [3] N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25 (2015), 1663–1763.
  * [4] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Advanced Mathematics and Applications, 9 (1999), 347-359.
  * [5] T. Black, Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity, Discrete Contin. Dyn. Syst. Ser. S 13 (2020) 119-137.
  * [6] T. Black, J. Lankeit, M. Mizukami, On the weakly competitive case in a two-species chemotaxis model. IMA J Appl Math. 81 (2016), 860-876
  * [7] J. Cao, W. Wang, and H. Yu, Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, J. Math. Anal. Appl. 436 (2016), no. 1, 382-392.
  * [8] K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21(1) (2016), 81-102.
  * [9] K. Fujie, M. Winkler, and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal. 109 (2014), 56-71.
  * [10] K. Fujie, M. Winkler, and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal dependent sensitivity, Math. Methods Appl. Sci., 38(6) (2015), 1212-1224.
  * [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, Heidelberg, New York, 1977\.
  * [12] T. Hillen, and K. Painter, A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009), 183–217.
  * [13] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis. Jahresber DMV 106 (2004), 51–69.
  * [14] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state anal-ysis, and blow-up results for multi-species chemotaxis models in the presence of attractionand repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), pp. 231–270
  * [15] D. Horstmann and M. Winkler, Boundedness vs.blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
  * [16] T. B. Issa and R.B. Salako, Asymptotic dynamics in a two-species chemotaxis model with non-local terms, Discrete and Continuous Dynamical Systems - B 22 (2017) 10:3839-3874.
  * [17] T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 2, 926-973.
  * [18] T. B. Issa and W. Shen, Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, Journal of Dynamics and Differential Equations, 31 (2019), no. 4, 2305-2338.
  * [19] T. B. Issa and W. Shen, Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments Journal of Dynamics and Differential Equations, 31, (2019), 1839–1871.
  * [20] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
  * [21] E.F. Keller and L.A. Segel, Traveling bans of chemotactic bacteria: a theoretical analysis. J. Theor. Biol.  30 (1971), 377–380.
  * [22] H. I. Kurt and W. Shen, Finite-time blow-up prevention by logistic source in chemotaxis models with singular sensitivity in any dimensional setting, SIAM J. Math. Anal., 53(1), (2021), 973–1003.
  * [23] H. I. Kurt and W. Shen, Chemotaxis models with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions, to appear in Nonlinear Analysis: Real World Applications.
  * [24] K. Lin, C. Mu, and L. Wang, Boundedness in a two-species chemotaxis system. Math Methods Appl Sci. 38 (2015), 5085-5096.
  * [25] M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic–elliptic type. Math Meth Appl Sci.  41 (2018), 234–249.
  * [26] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxiscompetition model with signal-dependent sensitivity,Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278.
  * [27] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Advanced Mathematics and Applications, 8 (1998), 145-156.
  * [28] G. Ren, B. Liu, A new result for global solvability in a singular chemotaxis-growth system with indirect signal production, Journal of Differential Equations, 337 (2022), 363-394.
  * [29] C. Stinner, J.I. Tello, and M. Winkler, Competitive exclusion in a two species chemotaxismodel, J. Math. Biol., 68 (2014), pp. 1607–1626.
  * [30] J.I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with logistic source, Nonlinearity, 25 (2012), pp. 1413–1425.
  * [31] W. Zhang and P. Niu, Asymptotics in a two-species chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems - B, (2021), 26 (8) : 4281-4298.