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\TopicFR{Équations aux dérivées partielles}
\TopicEN{Partial differential equations}

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\title{Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source}
\alttitle{Limite globale des solutions d'un modèle de consommation de chimiotaxie avec motilité dépendante du signal et source logistique}

\author{\firstname{Khadijeh} \lastname{Baghaei}}

% \address{\textcolor{red}{Address is missing}}

\thanks{This research was supported by a grant from 
PIAIS (No. 1402-10108).}

\CDRGrant[PIAIS]{1402-10108}


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 \begin{abstract}
 This paper deals with the following chemotaxis system:
 \begin{equation*}
	\begin{cases}
	u_{t}=\nabla\cdot\big(\gamma(v) \,\nabla u-u \,\xi(v) \,\nabla v\big)+\mu\, u\,(1-u), & x\in\Omega, \ t>0, \\
	v_{t}=\Delta v-uv, & x\in\Omega, \ t>0,
	\end{cases}
	\end{equation*}
 under homogeneous Neumann boundary conditions
 in a bounded domain 
 $ \Omega \subset \mathbb{R}^{n}, n\geq 2,$ 
 with smooth boundary.
Here, the functions $\gamma(v)$ and
$\xi(v)$ are as:
\begin{equation*}
 \gamma(v)=(1+v)^{-k}\quad
 \mbox{and} \quad
 \xi(v)=-(1-\alpha)\,\gamma'(v),
\end{equation*}
where $k>0$ and $\alpha \in (0,1).$\\
 For the above system, we prove that 
 the corresponding initial boundary value problem admits a unique global
 classical solution which is
 uniformly-in-time bounded.
This result is obtained under some conditions on initial value $ v_{0}$ 
 and $\mu$ and without any restriction on $k$ and $\alpha.$
 The obtained result extends the recent results obtained for this problem.
 
\end{abstract}


\begin{altabstract}
 Cet article porte sur le système de chimiotaxie suivant :
 \begin{equation*}
	\begin{cases}
	u_{t}=\nabla\cdot\big(\gamma(v) \,\nabla u-u \,\xi(v) \,\nabla v\big)+\mu\, u\,(1-u), & x\in\Omega, \ t>0, \\
	v_{t}=\Delta v-uv, & x\in\Omega, \ t>0,
	\end{cases}
	\end{equation*}
 sous des conditions aux limites homogènes de Neumann dans un domaine borné
 $ \Omega \subset \mathbb{R}^{n}, n\geq 2,$ 
 avec une frontière lisse. Ici, les fonctions $\gamma(v)$ et
$\xi(v)$ sont les suivantes :
\begin{equation*}
 \gamma(v)=(1+v)^{-k}\quad
 \mbox{and} \quad
 \xi(v)=-(1-\alpha)\,\gamma'(v),
\end{equation*}
où $k>0$ and $\alpha \in (0,1).$\\
 Pour le système ci-dessus, nous prouvons que le problème de valeur limite initiale correspondant admet une unique solution classique globale qui est uniformément bornée en temps.
Ce résultat est obtenu sous certaines conditions sur la valeur initiale $ v_{0}$ et $\mu$ et sans restriction sur $k$ et $\alpha.$
Le résultat obtenu étend les résultats récents obtenus pour ce problème. 
\end{altabstract}


\dateposted{2024-11-05}
\begin{document}
\maketitle


\section{Introduction}
In this paper, we consider the following initial boundary value problem:
\begin{equation}\label{yazahraadrecni}
	\begin{cases}
	u_{t}=\nabla\cdot\big(\gamma(v)\, \nabla u-u \,\xi(v)\, \nabla v\big)+\mu \, u\,(1-u), & x\in\Omega, \ t>0, \\
	v_{t}=\Delta v-uv, & x\in\Omega, \ t>0,\\
	 \frac{\partial u}{\partial \nu}= \frac{\partial v}{\partial \nu}= 0,& x\in \partial\Omega, \ t>0, \\
	u (x , 0)= u_{0}, \ v(x ,0)= v_{0}, & x\in\Omega,
	\end{cases}
	\end{equation}
where $ \Omega \subset \mathbb{R}^{n}, n\geq 2,$ is a bounded domain with smooth boundary, $\nu$ denotes the unit outward normal vector to $\partial \Omega$ and $u_{0}$ and $v_{0}$ are initial functions. In the above problem,
$u=u(x, t)$ is the cell density and $v=v(x, t)$ denotes the nutrient consumed chemical concentrations.\\
In mathematical biology, systems such as~\eqref{yazahraadrecni} describe the mechanism of chemotaxis. The chemotaxis is the movement of cells towards a higher concentration
a chemical signal substance produced by cells.
We first state the results related to the classical chemotaxis system, which has been introduced by Keller and Segel in~\cite{ks}. The classical chemotaxis system can be written as follows:
\begin{equation}\label{145}
	\begin{cases}
	u_{t}=\nabla\cdot\big(\gamma(v) \,\nabla u-u \,\xi(v)\, \nabla v\big)+\mu\, u\,(1-u), & x\in\Omega, \ t>0, \\
	\tau v_{t}= \Delta v - v + u, & x\in\Omega, \ t>0, 
	\end{cases}
	\end{equation}
 where $\tau \in \{0,1\}.$
 In the following, we write some important results for this problem in the absence and presence of the logistic source, respectively. In the absence of logistic source,
when $\gamma \in C^{3}((0, \infty))$ and 
$ \xi=-\gamma'$ as well as
\begin{align*}
 \limsup _{s \rightarrow \infty} \gamma(s) < \frac{1}{\tau},
\end{align*}
then the problem~\eqref{145} admits a unique global classical solution in any dimension~\cite{24}. Also, for $n \geq 1,$
when the function $\gamma$ has strictly positive
 upper and lower bounds, then the classical solutions are 
 uniformly-in-time bounded~\cite{24}. 
 This result is also proved
 when $n \geq 2$ and the function $\gamma$
 decays at a certain slow rate
at infinity~\cite{24}. 
Among the special functions that have been studied as $\gamma$ are: 
\begin{align*}
 & \gamma(v)= c_{0} \, v^{-k}\quad \mbox{with} \quad k>0 \quad \mbox{and} \quad c_{0} >0
 \end{align*}
 and
\begin{align*} 
 & \gamma(v)=\expe^{-\chi v}\quad \mbox{with} \quad \chi>0.
\end{align*}
For $\gamma(v)= c_{0} \, v^{-k},$ 
 for all $k>0,$ 
 the global existence and boundedness of the solution is proved
 under a smallness assumption on $c_{0}$ in any dimension~\cite{Yk}.
 Also, for this function, by removing the assumption on $c_{0},$ 
 when $n \geq 2$ and $k \in (0, \frac{2}{n-2}),$ the same result is proved in cases $\tau=0$ 
~\cite{AY} and $\tau=1$~\cite{FS}.
 In the other special case $\gamma(v)= \expe^{-\chi v},$ if $n=2$ and
 $\int_{\Omega} u_{0} \, \dd x < \frac{4 \pi}{\chi},$ then the classical solutions are global and bounded whereas for $\int_{\Omega} u_{0} \, \dd x > \frac{4 \pi}{\chi},$
 blow up occurs either in finite or infinite time~\cite{2020}. But, in the case of
 $\tau=0,$
 the blow up occurs in infinite time~\cite{Fj}.
 Also, for $n=2,$ the classical
solutions are globally bounded if the positive function $\gamma$ decreases slower than an exponential
speed at high signal concentrations~\cite{2021}. For $n \geq 3,$ the same result is true when
$\gamma$ decreases at certain algebraically speed~\cite{2021}.
We now state the result obtained in
\cite{20199} which is the global existence of very weak solution to the problem~\eqref{145} when $\gamma(v)=\frac{1}{c+v^{k}}$ with $c\geq 0$ and $k>0.$ This result was obtained without any smallness assumption on the initial data provided that
\begin{equation*}
	 k\in 
	 	\begin{cases}
	 (0, \frac{7}{3}), &\mbox{if } n=1, \\
	(0,2), &\mbox{if } n=2,\\
(0,\frac{4}{3}), & \mbox{if } n=3.
	\end{cases}
	\end{equation*}
 Also, for $n\geq 1$ with $\gamma(v)=\expe^{-v},$ the existence of weak-strong solutions is proved 
 in
~\cite{2021_1}.
We now state the results to the problem~\eqref{145} in the presence of logistic source.
 If the decreasing function $\gamma \in C^{3}([0, \infty))$ satisfies
$\lim _{s \rightarrow \infty}\gamma(s)=0 $
and $\lim _{s \rightarrow \infty}\frac{\gamma'(s)}{\gamma(s)} $ exists, then the classical solutions are global and bounded for $n=2$~\cite{55}.
For $n \geq 3,$ the same result is true when $\mu >0$ is large and the last condition is replaced with $|\gamma'(s)| \leq m,$ where $m$ is some positive constant~\cite{2019}.
Also, in two dimensional case, the same result holds when $\gamma \in C^{3}((0, \infty))$ 
and $|\,\gamma'\,|+|\,\gamma''\,| \in L^{\infty}((0, \infty))$
~\cite{winkler2022}.
\\
Now,
 we write some important results related to the problem~\eqref{yazahraadrecni}.
The origin of the definition of this problem comes from 
the following chemotaxis-Navier–Stokes system which describes the motion of oxygen-driven swimming bacteria in an in-compressible
 fluid 
\begin{equation*}
	\begin{cases}
	u_{t}+ \omega \cdot \nabla u =\nabla\cdot\big( \nabla u-u \,\xi(v) \nabla v\big), & x\in\Omega, \ t>0, \\
	v_{t}+ \omega \cdot \nabla v=\Delta v-ug(v), & x\in\Omega, \ t>0,\\
	\omega_{t}+ (\omega \cdot \nabla) \omega =\Delta \omega -\nabla P+ u \nabla \phi, & x\in\Omega, \ t>0, \ t>0, \\
 \nabla \cdot \omega=0, & x\in\Omega, \ t>0, \ t>0. \\
	\end{cases}
	\end{equation*}
 Here, $u$ denotes the bacteria density and $v$ is the oxygen concentration. Also, $\omega$ and $P$ are the velocity and pressure of the
fluid, respectively. The function $ \xi$ measures the chemotactic sensitivity, $g$ is the consumption rate of the oxygen by the bacteria, and $\phi$
is a given potential function~\cite{2005}.
For the related results
with the chemotaxis-Navier–Stokes systems, we refer the interested readers to~\cite{2021w, 2014w,2016l, 2023} and references therein. We see that 
the problem~\eqref{yazahraadrecni} can be obtained from the preceding chemotaxis-Navier–Stokes system in the case of $\gamma(v)\equiv 1$ with the choice $\omega\equiv 0$ and $ g(v)=v.$
 For the problem~\eqref{yazahraadrecni},
in the absence of logistic source, when $\gamma(v) \equiv 1,\, \xi(v) \equiv \chi,$
where $\chi$ is some positive constant, for $n=2,$ the classical solutions are global and bounded in bounded convex domains with a smooth boundary~\cite{taowin}.
Also, for $n \geq 3,$ this result holds in bounded domains with a smooth boundary provided that
$
\|v_{0}\|_{L^{\infty}(\Omega)}\leq \frac{1}{6\,(n+1)\chi}$~\cite{tao}.
Later, this condition was extended to 
$\|v_{0}\|_{L^{\infty}(\Omega)}\leq \frac{\pi}{\chi\, \sqrt{2(n+1)}}$ 
and the same result was obtained in the absence of logistic source~\cite{baghaei-khelghati} and the presence of logistic source
~\cite{baghaei-khelghati-logistic}.\\
We now state the related results to the
problem~\eqref{yazahraadrecni}
in the absence of logistic source,
when the function $\gamma$ is not constant.
We begin by stating the results obtained in~\cite{GW} which is for the positive function $\gamma \in C^{0}([0, \infty))$
and
$ \xi=-\gamma'.$ For non-negative initial data from
 $(C^{0}(\overline{\Omega}))^{*} \times L^{\infty}(\Omega)$, this problem admits a global very weak solution in all dimensions, also, 
 for $\gamma \in C^{1}([0, \infty)),$ 
 the solutions stabilize toward a semi-trivial spatially homogeneous steady
state in the large time limit.
This result is obtained for $n\leq 3$ in~\cite{GW}
 and later for $n \geq 1$ in~\cite{ph}.
 If the decreasing function $\gamma $ belongs to $ C^{3}([0, \infty))$ and 
$ \xi=-\gamma',$ then for non-negative initial data from $C^{0}(\overline{\Omega}) \times W^{1,\infty}(\overline{\Omega})$ under a smallness assumption on $\|v_{0}\|_{L^{\infty}(\Omega)},$ there exists a unique global classical solution 
 that is bounded 
~\cite{6}.
If the initial data $(u_{0}, v_{0})$ belongs to $(W^{1,\infty}(\Omega))^{2}$ and $\gamma \in C^{3}([0, \infty)),$
 then classical solutions are globally bounded when $n\leq 2,$ and weak solutions are global
when $n \geq 3,$ in particular, such weak solutions become eventually smooth if $n = 3$~\cite{app}.
If $\gamma\in C^{0}([0, \infty))\cap C^{3}([0, \infty)),$
$\gamma(s)>0$
for all $s > 0$ as well as
$ \xi=-\gamma',$ and 
$\gamma$ satisfies:
\begin{equation*}
 \liminf_{s \searrow 0}
 \frac{\gamma(s)}{s^{\alpha}}>0
 \quad \mbox{and}
 \quad \limsup_{s \searrow 0} s^{\beta} \,|\gamma'(s)|< \infty
\end{equation*}
with $\alpha > 0$ and $\beta > 0,$
then this problem admits a global generalized solution for all reasonably regular initial data~\cite{w}. 
In~\cite{tw2023}, 
the function $\gamma$ is assumed to satisfy:
\begin{equation*}
 c_{1}\, s^{-k} \leq \gamma(s) \leq c_{2}\, s^{-k}
 \quad \mbox{for all} \quad s>0, \quad (*)
\end{equation*}
where $k,$ $c_{1}$ and $c_{2} $ are some positive constants. Under this assumption for
$\gamma \in C^{3}([0, \infty))$
and the initial data belongs to $(W^{1,\infty}(\Omega))^{2},$
the problem~\eqref{yazahraadrecni}
admits a global classical solution when $n=1,$ and a global weak-strong solution
when $n \geq 2.$ Also, this result is true if $2\leq n\leq 5,$ $k>\frac{n-2}{6-n}$ and
$\gamma$ satisfies $(*)$ and
\begin{equation*}
 |\gamma'(s)| \leq c_{3}\, s^{-k-1}
 \quad \mbox{for all} \quad s>0,
\end{equation*}
where $c_{3} $ is some positive constant~\cite{tw2023}.\\
Finally, we state the important results related to the
problem~\eqref{yazahraadrecni}
in the presence of logistic source.
If $\gamma \in C^{3}([0, \infty))$ and $\gamma'(s)<0$
for all $s \geq 0$ as well as
$ \xi=-\gamma',$ then
for $n = 2$ and $\mu >0,$ the solutions are global and bounded~\cite{5}. Also, the same result is obtained when $n \geq 3$ and $\mu$ is suitably large~\cite{5}. Moreover,
 the solution converges exponentially to $(1, 0)$ when $t$ tends to infinity~\cite{5}. For this problem,
 when the logistic source is as $f(u)=a\, u -\mu \,u^{\kappa}$ with $a>0, \, \mu>0$ and $\kappa>1,$ the classical solutions are global and bounded
 if one of the cases 
 $\big( n\leq 2, \kappa>1;$
 $ n\geq 3, \kappa>2$ or
 $ n\geq 3, \kappa=2$ and $\mu$ is large\big) holds~\cite{7}.
If the positive decreasing function $\gamma$ belongs to $ C^{2}([0, \infty))$ and
 $\gamma''\geq 0$ as well as $ \xi=-(1-\alpha)\,\gamma'$
 with $\alpha \in (0,1),$ in~\cite{4}, it is proved that the problem~\eqref{yazahraadrecni}
 admits a unique global classical solution that is uniformly
in time bounded provided that:
 \begin{align*}
 & \frac{(\gamma'(s))^{2}}{\gamma''(s)}
 \leq \frac{n}{2(n+1)^{3}}, \qquad 0< \|v_{0}\|_{L^{\infty}(\Omega)}\leq \gamma^{-1}\left(\frac{1}{n+1}\right)
 \end{align*} 
 and
 \begin{align*}
 \mu > \max_{0< s \leq \|v_{0}\|_{L^{\infty}(\Omega)}}
 \frac{-\gamma'(s)\, \|v_{0}\|_{L^{\infty}(\Omega)}}{\gamma(s)}.
 \end{align*}
 We see that the above conditions in the case of $\gamma(s)=(1+s)^{-k}\,(k>0)$ are as follows:
\begin{align*}
 k<\frac{n}{2(n+1)^{3}-n},
 \quad \quad
 \mu>k\|v_{0}\|_{L^{\infty}(\Omega)}
\qquad \mbox{and} \qquad 
 \|v_{0}\|_{L^{\infty}(\Omega)}\leq \gamma^{-1}\left(\frac{1}{n+1}\right). 
\end{align*}
Because of $\gamma'<0,$
the last condition is written as:
\begin{align*}
0< \|v_{0}\|_{L^{\infty}(\Omega)}\leq (n+1)^{\frac{1}{k}}-1.
\end{align*}
In~\cite{KHB}, we studied the special case
$\gamma(s)=(1+s)^{-k}\,(k>0)$ 
and we were able to improve the conditions in~\cite{4}. In fact, we proved the same result in~\cite{4} under the following conditions:
 \begin{align*}
k\,(1-\alpha)<\frac{4}{n+5}, \qquad 0<\|v_{0}\|_{L^{\infty}(\Omega)}\leq \Bigg[ \frac{4\big[l-1\big]}{ n+1}\Bigg]^{\frac{1}{k}}-1
 \end{align*}
and
\begin{equation*}
 \mu>
 \frac{n\|v_{0}\|_{L^{\infty}(\Omega)}}{l\,(n+1)(1+\|v_{0}\|_{L^{\infty}(\Omega)})}
 \end{equation*}
 with $l=\frac{1}{k\,(1-\alpha)}.$
In this paper, we focus again on the functions $\gamma$ and
$\xi$ as follows:
\begin{equation}\label{yarahim+}
 \gamma(s)=(1+s)^{-k}\quad
 \mbox{and} \quad
 \xi(s)=-(1-\alpha)\gamma'(s),
\end{equation}
where $k>0$ and $\alpha \in (0,1)$ 
 and extend our recent result. 
In fact, we remove the condition on $ k\,(1-\alpha)$
and prove the solutions are uniformly in time bounded under some conditions on $ \|v_{0}\|_{L^{\infty}(\Omega)}$ and $ \mu.$

\section{Our results}

Here, we state the standard well-posedness and classical solvability result.
\begin{lemma}\label{lem3}
 Let $u_{0} \geq 0$ and $v_{0}\geq 0$ satisfy $(u_{0}, v_{0} ) \in (W^{1,r}(\Omega))^{2}$ for some $r > n.$ Then problem~\eqref{yazahraadrecni} has a unique local in time
classical solution 
\begin{equation*}
 (u,v) \in \left(C \big([0, T_{\max}); W^{1,r}(\Omega)\big) \cap C^{2,1}\big(\,\overline{\Omega}\times (0, T_{\max})\right)^{2}
\end{equation*}
where $T_{\max} $ denotes the maximal existence time. In addition, if $T_{\max}< + \infty, $
then:
\begin{equation*}
 \limsup_{t\rightarrow T_{\max} }
 \|u(\ .\ ,t)\|_{L^{\infty}(\Omega)}= + \infty.
\end{equation*}
 Moreover, $u$ and $v$ satisfy the following inequalities:
\begin{equation}\label{yaallah1}
 u \geq 0 \quad \mbox{and} \quad 0 \leq v \leq \|v_{0}\|_{L^{\infty}(\Omega)}
 \quad \mbox{in} \quad \Omega\times (0, T_{\max}),
\end{equation}
also,
\begin{align}\label{mamnonam}
 \int_{\Omega} u(\ .\ , t) \, \dd x \leq c,
\end{align}
where $c$ is some positive constant.
\end{lemma}
For details of the proof, we refer the reader to~\cite{55, 4}.\\
Based on main ideas in~\cite{baghaei-khelghati,
baghaei-khelghati-logistic, KB1, KB2}, we write the following 
key lemma similar to~\cite[Lemma~2.2]{KHB}.
\begin{lemma}\label{lem1}
	Let $(u, v)$ be the solution of problem~\eqref{yazahraadrecni}. If there exists a smooth
positive function $\varphi(s)$ such that for
$0\leq s\leq \|v_{0}\|_{L^{\infty}(\Omega)},$
 the following inequality holds: 
\begin{equation}\label{yaallah}
(B(s))^{2}-4\, A(s)\, C(s) \leq 0,
\end{equation}
where for $p\geq 2,$ the functions $A, B$ and $C$ are defined as:
 \begin{equation}\label{313}
	\begin{cases}
	 A(s)=(p-1)\, \varphi (s)\, \gamma(s), \\
	B(s)=(p-1)\, \varphi (s)\, \xi(s) -\varphi' (s) \, (\gamma(s)+1), \\
 C(s)= \frac{1}{p}\, \varphi ''(s)- \varphi' ( s)\, \xi(s),
	\end{cases}
	\end{equation}
 then:
	\begin{equation*}
	 \frac{1}{p}\,\frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x 
 \leq -\int _{\Omega} \left[\mu\, \varphi (v)+\frac{1}{p}\, v \, \varphi' (v)\right] u^{p+1}\, \dd x
 +\mu \int_{\Omega}u^{p} \varphi (v)\, \dd x.
	\end{equation*}
 \begin{proof}
 We assume that there exists
a smooth
positive function $\varphi(s)$ such that
for $0\leq s\leq \|v_{0}\|_{L^{\infty}(\Omega)}$ and $p\geq 2,$~\eqref{yaallah} holds. We take this function
and
 use~\eqref{yazahraadrecni} and integration by parts to write:
\begin{equation}
\label{yasahebazzaman} 
\begin{aligned}
 \frac{1}{p}\frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x & =
 \int _{\Omega} u^{p-1} \varphi ( v)\, u_{t} \, \dd x
 + \frac{1}{p}\,\int_{\Omega}u^{p} \, \varphi' ( v) \, v_{t} \, \dd x %\nonumber
 \\
 &=
 - (p-1) \int_{\Omega} u^{p-2} \varphi ( v) \,\gamma(v) \, |\nabla u|^{2} \, \dd x
 %\nonumber
%\end{aligned} 
 \\
%\begin{aligned}
 & \hspace*{15mm}+ \int _{\Omega}u^{p-1} \left[\, (p-1)\, \varphi (v) \,\xi(v) -\varphi' (v) \, (\gamma(v)+1) \right]
 \big(\nabla u \cdot \nabla v\big) \, \dd x
 %\nonumber
 \\
 & \hspace*{15mm}+ \int _{\Omega}u^{p} \left[\, \varphi' ( v) \,\xi(v) - \frac{1}{p} \,\varphi ''(v ) \right] \,|\nabla v |^{2} \, \dd x 
 %\nonumber
 \\
 & \hspace*{15mm}- \int _{\Omega} \left[\,\mu\, \varphi (v)+\frac{1}{p} \,v \, \varphi' (v)\right] u^{p+1}\, \dd x
 +\mu \,\int_{\Omega}u^{p} \varphi ( v)\, \dd x.
 \end{aligned}
 \end{equation}
 For convenience in calculations, we write~\eqref{yasahebazzaman} as follows:
	\begin{equation}\label{yarahim}
	 \frac{1}{p}\,\frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x 
	= \int _{\Omega} J (u, v) \, \dd x 
	- \int _{\Omega} \left[\,\mu\,\varphi (v)+\frac{1}{p} \, v \, \varphi' (v)\right] u^{p+1}\, \dd x
 +\mu \int_{\Omega} u^{p} \varphi ( v)\, \dd x
	\end{equation}
	with
\begin{equation}
	\begin{aligned}\label{ya mahdi adrecni}
	J(u, v)&=
 - (p-1)\, u^{p-2} \varphi (v)\, \gamma(v) \, |\nabla u|^{2} %\nonumber
 \\
 & \hspace*{15mm}
 +u^{p-1} \left[\, (p-1) \,\varphi (v) \,\xi(v) \,-\varphi' (v) \, (\gamma(v)+1) \right]
 \big(\nabla u \cdot \nabla v\big) 
 %\nonumber
 \\
 & \hspace*{15mm}
 + u^{p} \left[\, \varphi' ( v)\, \xi(v) - \frac{1}{p} \, \varphi ''(v ) \right] |\nabla v |^{2} %\nonumber
 \\
&=	 - u^{p-2} \, |\nabla u |^{2} A(v)
	+u^{p-1} 
	\big( \nabla u \cdot
	\nabla v \big) B(v)
	- u^{p} \,|\nabla v |^{2} \, C(v),
	\end{aligned}
	\end{equation} 
	where $ A, B $ and $ C $ are defined in~\eqref{313}.
	Now, by considering~\eqref{ya mahdi adrecni}, we can write
	\begin{align*}
J(u,v) 
& =-\bigg(\sqrt{ u^{p-2} A(v)}\, \nabla u
- \frac{u^{p-1}B(v)}{2\sqrt{ u^{p-2} A(v)}}\, \nabla v\bigg)\cdot\bigg(\sqrt{ u^{p-2} A(v)}\, \nabla u
-\frac{u^{p-1}B(v)}{2\sqrt{ u^{p-2} A(v)}}\, \nabla v\bigg)
\\
&\hspace*{78mm}+u^{p}\left[ \frac{(B(v))^{2}}{4 \, A(v)}-C(v)\right]|\nabla v |^{2}\quad \
 \\
&\leq 
u^{p}\left[ \frac{(B(v))^{2}-4\,A( v)\, C(v)}{4 \,A(v)} \right]|\nabla v |^{2}.
\end{align*}
	In view of the condition~\eqref{yaallah}, we see that 
$J\leq 0.$ Thus, the equality~\eqref{yarahim}
becomes
	\begin{equation*}
	 \frac{1}{p}\, \frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x 
\leq -\int _{\Omega} \left[\,\mu\, \varphi (v)+\frac{1}{p}\, v \, \varphi' (v)\right] u^{p+1}\, \dd x +
 \mu \int_{\Omega}u^{p} \varphi ( v)\, \dd x.
	\end{equation*}
This completes our proof.
 \end{proof}
 \end{lemma}
 Before presenting a smooth positive function
 $\varphi$ such that the relation~\eqref{yaallah} holds,
 we present the following preliminary lemma.
\begin{lemma}\label{lemk}
Let the function $\gamma$ for $s\geq 0$ is defined as $\gamma(s)=(1+s)^{-k}$ with $k>0.$
If the function $H$ for $z>0$ is defined as
 \begin{equation}\label{1212}
 H(z) 
 = 
d^{2}\,p\,z^{2\lambda}-2\, d ^{2}(p-2)\, z^{2\lambda-1} +d^{2}p\,z^{2\lambda-2} %\\
 - 2\, d \big[2\, d \,(p-1)-p
 +2 \,l\big]z^{\lambda-1}
 -2\, d \, p\,z^{\lambda} +p,
\end{equation}
where for $\alpha \in (0,1)$ and $p>2,$ 
the parameters $\lambda, l$ and $d$ are as follows:
\begin{equation*}
 \lambda=d\,(p-1)\,(1-\alpha),\qquad l=\frac{1}{k\,(1-\alpha)} 
\end{equation*}
and
 \begin{align}\label{3132}
 d
 >\frac{-l+\sqrt{\,l^{2}+p\,(p-2)}}{2\,(p-2)}.
\end{align} 
Then there exists $\delta_{0}>0$
such that for 
$0<s\leq \delta_{0},$ $H(\gamma(s))\leq 0.$
\begin{proof}
 At first, we see that 
 \begin{align*}
 H(1) 
 &= 
d^{2}\,p-2\, d ^{2}(p-2) +d^{2}\,p
%\nonumber\\
% & \quad
 - 2\, d \big[2\, \dd \,(p-1)-p
 +2 \,l\big]
 -2\, d \,p+ p\\
 & =
- 4\,(p-2)\, d ^{2}-4 \,l\, d +p.
\end{align*}
It is not difficult to see that the choice of $d$ as 
\eqref{3132}
implies that:
\begin{align*}
- 4\,(p-2)\, d ^{2}-4 \,l\, d +p
 <0.
\end{align*}
Thus, $ H(1)<0.$
By considering
the continuity of the function $H$ on $(0, \infty)$ and $\gamma$ on $[0, \infty),$ and also
$ H(\gamma(0))=H(1)<0,$
we conclude that there exists $\delta_{0}>0$
such that for 
$0<s\leq \delta_{0},$ $H(\gamma(s))\leq 0.$
\end{proof}
\end{lemma}
 We now present a smooth positive function
 $\varphi$ and show that for this function, the relation~\eqref{yaallah} holds.
We must state that
the following lemma is the only place where the special choice of $\gamma$ is used.
\begin{lemma}\label{lem 2}
Assume that the initial values $u_{0}$
 and $ v_{0} $ are non-negative and
 satisfy: 
 \begin{equation*}
 (u_{0}, v_{0} ) \in (W^{1,r}(\Omega))^{2} 
 \quad \mbox{for some} \quad r > n,
 \end{equation*}
 and 
 \begin{equation*}
 0<\|v_{0}\|_{L^{\infty}(\Omega)}\leq \delta_{0},
 \end{equation*}
 which $\delta_{0}$ is introduced in Lemma~\ref{lemk}.
Also, assume that the functions $\gamma(v)$
and $\xi(v)$
	are defined as~\eqref{yarahim+}
 and
 for $p>n,$ the following condition
 holds:
 \begin{equation}
 \label{sallamkhoda}
 \mu>
 \frac{d\, (p-1)\, \|v_{0}\|_{L^{\infty}(\Omega)}}{l\,p\,(1+\|v_{0}\|_{L^{\infty}(\Omega)})},
 \end{equation}
 where $l=\frac{1}{k\,(1-\alpha)}$ 
 and $d$ is chosen as~\eqref{3132}.
Then, there exists some positive constant $c$ which also depends on $\delta_{0}$ and $p$ such that the following estimate holds:
\begin{equation}\label{313*}
\|u(\ .\ ,t)\|_{L^{p}(\Omega)}\leq c, \qquad \forall \,t \in (0, T_{\max}).
\end{equation}
 \begin{proof}
We prove this lemma in two steps. In the first step, we present a smooth positive function $\varphi$ and show that for this function~\eqref{yaallah} holds. 
In the second step, we prove~\eqref{313*} holds.


\begin{proof}[Step 1] 
 As a starting point, in Lemma~\ref{lem1}, we take
 $p > n$ and define the smooth positive function $\varphi$ as follows:
\begin{equation*}
 \varphi ( v)=\expe^{(\gamma(v))^{\lambda}}
\end{equation*} 
with $ \lambda=d\,(p-1)\,(1-\alpha),$
where $d$ is taken as~\eqref{3132}.
For this function, we have:
\begin{align}
	&\varphi'(v)
=\lambda \, (\gamma(v))^{\lambda-1}\gamma'(v) \,\varphi ( v)\label{127}
\end{align}
and
\begin{align*}
	&\varphi''(v)
=\lambda\,\big[\,(\lambda-1) (\gamma'(v))^{2} + \gamma(v)\gamma''(v) 
+\lambda \, (\gamma(v))^{\lambda} (\gamma'(v))^{2} \,\big](\gamma(v))^{\lambda-2}\varphi ( v).\nonumber
	\end{align*}
 Because of 
 \begin{align*}
 \gamma'(v)=-k\,(1+v)^{-k-1}
 \quad \mbox{and}
 \quad 
 \gamma''(v)=k\,(k+1)\,(1+v)^{-k-2},
 \end{align*}
 we see that $ \gamma(v) \gamma''(v)=\big(1+\frac{1}{k}\big)(\gamma'(v))^{2}.$
 Thus,
 \begin{equation}
	\varphi''(v)
=\lambda\left[\lambda+\frac{1}{k}
+\lambda \,(\gamma(v))^{\lambda} \right](\gamma(v))^{\lambda-2}\,(\gamma'(v))^{2} \varphi ( v).\label{125}
	\end{equation}
 By considering the values of $A, B$ and $C$ from~\eqref{313}, we can write:
\begin{multline*}
 (B(v))^{2}-4 \,A( v) \,C(v) %\nonumber\\
 =(p-1)^{2} (\varphi(v)) ^{2} \, (\xi (v)) ^{2}+
 (\varphi' (v)) ^{2} \, (\gamma(v)+1)^{2} \nonumber
 \\
 \quad
 -\frac{4 \,(p-1)}{p}\, \varphi ( v) \, \varphi ''(v )\, \gamma(v)
 -2\,(p-1)\,\varphi ( v)\, \varphi' ( v)\, \xi(v) \, (1-\gamma(v)).
\end{multline*}
Making use of~\eqref{127},~\eqref{125} and $\xi(v)=-(1-\alpha)\gamma'(v),$
we obtain:
\begin{multline*}
 (B(v))^{2}-4\, A( v)\, C(v)%\\
 =(\gamma'(v))^{2}(\varphi(v)) ^{2}\left\{(p-1)^{2}(1-\alpha)^{2} +
 \lambda^{2} (\gamma(v))^{2\lambda-2} (\gamma(v)+1)^{2}
 \vphantom{\frac{4\,\lambda\, (p-1)}{p}}\right.\\
 %\quad
 \left. -\frac{4\,\lambda\, (p-1)}{p} \,\left[\lambda+\frac{1}{k}
+\lambda (\gamma(v))^{\lambda} \right](\gamma(v))^{\lambda-1}\, 
%\\& \quad
 +2\,\lambda\,(p-1)(1-\alpha) (\gamma(v))^{\lambda-1} (1-\gamma(v))\right\}.
 \end{multline*}
 Because of $ \lambda=d\,(p-1)(1-\alpha),$
 we replace $(p-1)(1-\alpha)$ with $\frac{\lambda}{d}$
 to have:
 \begin{multline}\label{11012}
 (B(v))^{2}-4\, A( v)\, C(v)\\
 \begin{aligned}
 {}={}&(\gamma'(v))^{2}(\varphi(v)) ^{2}\Bigg\{\frac{\lambda^{2}}{d^{2}} +
 \lambda^{2} (\gamma(v))^{2\lambda-2} (\gamma(v)+1)^{2}\\
 & \quad
 -\frac{4\,\lambda^{2}}{d\,p\,(1-\alpha)}
 \,\left[\lambda \, \big(1+(\gamma(v))^{\lambda}\big)+\frac{1}{k}
 \right](\gamma(v))^{\lambda-1}
 +\frac{ 2\,\lambda^{2} }{d} (\gamma(v))^{\lambda-1} (1-\gamma(v))\Bigg\}
\\
 {}={}&\frac{\lambda^{2}(\gamma'(v))^{2}(\varphi(v)) ^{2} }{d^{2}\, p}\Bigg\{ p + d^{2}\, p\,
 (\gamma(v))^{2\lambda-2} (\gamma(v)+1)^{2}\\
 & \quad 
 -4\, \dd \,\left[d\,(p-1)(1+(\gamma(v))^{\lambda})+\frac{1}{k\,(1-\alpha)} \right](\gamma(v))^{\lambda-1}
 +2\, \dd \,p\, (\gamma(v))^{\lambda-1} (1-\gamma(v))\Bigg\}\, \\
 {}={}&\frac{\lambda^{2} (\gamma'(v))^{2}(\varphi(v)) ^{2}}{d^{2}\, p}\Bigg\{p +
\, \dd ^{2}p\,(\gamma(v))^{2\lambda}-2\, \dd ^{2}(p-2) (\gamma(v))^{2\lambda-1} \\
 & \quad
 +d^{2}\,p\,(\gamma(v))^{2\lambda-2} - 2\, \dd \big[2\, \dd \,(p-1)-p
 +2 \,l\big](\gamma(v))^{\lambda-1}
 -2\, \dd \,p\,(\gamma(v))^{\lambda}\Bigg\}\\
 {}\coloneqq{} &\frac{\lambda^{2}}{d^{2}\, p}(\gamma'(v))^{2} (\varphi(v)) ^{2}
 H(\gamma (v)),
\end{aligned}
\end{multline}
where the function $H$ is defined by~\eqref{1212}.
By considering Lemma~\ref{lemk},
we see that there exists $\delta_{0}>0$
such that for 
$0<v\leq \delta_{0},$ $H(\gamma(v))\leq 0.$ Because of
$0<\|v_{0}\|_{L^{\infty}(\Omega)}\leq \delta_{0},$ we conclude that $H(\gamma(v))$ is non-positive in the interval $(0,\|v_{0}\|_{L^{\infty}(\Omega)}]. $ 
 Therefore, by
combining this with~\eqref{11012}, 
 we conclude that
 the relation~\eqref{yaallah} holds. 
\let\qed\relax
\end{proof}
\begin{proof}[Step 2]
 In order to prove~\eqref{313*}, we apply Lemma~\ref{lem1} and write:
\begin{multline}\label{sallamfrmandeh}
	 \frac{1}{p}\,\frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x +\mu \int_{\Omega}u^{p} \varphi ( v)\, \dd x %\nonumber
	 \\
 \leq -\int _{\Omega} \left[\mu\, \varphi (v)+\frac{1}{p}\, v \, \varphi' (v)\right] u^{p+1}\, \dd x +2\,\mu \int_{\Omega}u^{p} \varphi ( v)\, \dd x.
	\end{multline}
We now apply the Young inequality to the second term on the right hand side of~\eqref{sallamfrmandeh} to have: 
 \begin{align*}
 2\,\mu \int_{\Omega}u^{p} \, \varphi ( v)\, \dd x 
 &
 \leq \epsilon \int_{\Omega}u^{p+1} \varphi ( v)\, \dd x\, 
 + c_{\epsilon} \int_{\Omega}\varphi ( v)\, \dd x, 
 \end{align*}
 where $\epsilon$ is chosen as follows:
 \begin{equation}\label{yamahdi}
 0< \epsilon <\mu-
 \frac{ k\,\lambda \,\|v_{0}\|_{L^{\infty}(\Omega)}}{p\, (1+\|v_{0}\|_{L^{\infty}(\Omega)})}
 \end{equation}
 and:
 \begin{equation*}
 c_{\epsilon}=\frac{1}{p+1}\,\left[\frac{p}{\epsilon\,(p+1)}\right]^{p}(2\mu)^{p+1}.
 \end{equation*}
 Because of $\gamma'(v)<0,$~\eqref{127} implies that $\varphi' ( v)<0.$ Thus, 
 \begin{align}\label{yaali}
 \varphi (\|v_{0}\|_{L^{\infty}(\Omega)}) < \varphi ( v)<\varphi (0).
 \end{align}
 Making use of this, we can write:
 \begin{align}\label{shokr}
 2\,\mu \int_{\Omega}u^{p} \varphi ( v)\, \dd x 
 &
 \leq \epsilon \int_{\Omega}u^{p+1} \varphi ( v)\, \dd x+ c_{0} 
 \end{align}
 with $c_{0} =c_{\epsilon}\,|\Omega| \,\varphi (0).$
 We now combine the inequality~\eqref{shokr}
 with~\eqref{sallamfrmandeh}
 to obtain:
 \begin{equation}\label{khodayakomakamkon}
	\frac{1}{p}\, \frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x +\mu \int_{\Omega} u^{p} \varphi ( v)\, \dd x %\nonumber
%	\\
 \leq \int _{\Omega} \left[(\epsilon-\mu) \,\varphi (v)-\frac{1}{p}\, v \, \varphi' (v)\right] u^{p+1}\, \dd x
+ c_{0}.
	\end{equation}
We use $\gamma'(v)=-\frac{k}{1+v} \gamma(v), \, \gamma(v)\leq 1$ and~\eqref{yamahdi} to have
 \begin{align*}
 (\epsilon-\mu)\, \varphi (v)-\frac{1}{p} v \, \varphi' (v)
 &=\left[
 \epsilon-\mu-\frac{ \lambda }{p}\, v \, (\gamma(v))^{\lambda-1}\gamma'(v)\right] \varphi ( v)\\
 &=
 \left[
 \epsilon-\mu+\frac{ k\,\lambda\, v }{p\,(1+v)}\, \, (\gamma(v))^{\lambda}\right] \varphi ( v) \\
 & \leq
 \left[ \epsilon-\mu+\frac{ k\,\lambda \,\|v_{0}\|_{L^{\infty}(\Omega)}}{p\, (1+\|v_{0}\|_{L^{\infty}(\Omega)})}\, \right] \varphi ( v)\\
 & \leq 0.
 \end{align*} 
 This along with~\eqref{khodayakomakamkon} yields:
 \begin{equation*}
	 \frac{\dd}{\dd t}\int _{\Omega} u^{p} \varphi (v) \, \dd x +\mu\,p\int_{\Omega}u^{p} \varphi ( v)\, \dd x
 \leq
p\,c_{0}.
	\end{equation*}
 We put: 
 \begin{equation*}
 y(t)=\int _{\Omega} u^{p} \varphi (v) \, \dd x.
 \end{equation*}
 Thus,
 \begin{align*}
y'(t)
 +\mu\,p \,y(t)
 \leq
 p\,c_{0}.
	\end{align*}
 This yields:
 \begin{align}\label{14}
y(t)
 \leq \max\Big\{y(0), \frac{c_{0}}{\mu}
\Big\}.
	\end{align}
This along with~\eqref{yaali} allows us to write:
 \begin{equation*}
 \int _{\Omega} u^{p} \, \dd x \leq
 \big( \varphi (\|v_{0}\|_{L^{\infty}(\Omega)})\big)^{-1} \max\Big\{y(0), \frac{c_{0}}{\mu }
\Big\}\coloneqq c.
 \end{equation*}
By considering the value of $c_{0}, $ we see that $c$ also depends on $\delta_{0}$ and $p.$
 This completes our proof.
\end{proof}
\let\qed\relax
\end{proof}
\end{lemma}
 The proof of the following lemma is the same as~\cite[Lemma~3.2]{tao} or~\cite[Lemma~2.4]{KHB}. 
 But, we write it to complement our content.
 \begin{lemma}\label{lem5}
Assume that for $p>n,$ the following estimate
 holds:
\begin{equation}\label{77}
\|u(\ .\ ,t)\|_{L^{p}(\Omega)}\leq C_{1}, \qquad \forall \,t \in (0, T_{\max}),
\end{equation}
where $C_{1}$ is some positive constant, then there exists some positive constant $C_{2}$ which also depends on $C_{1}$ 
 and $
 \|v_{0}\|_{L^{\infty}(\Omega)}$ 
 such that
\begin{equation}\label{313*****}
 \|\nabla v(\ .\ ,t)\|_{L^{\infty}(\Omega)} \leq C_{2}
\end{equation}
for all $t \in (0, T_{\max}).$
\begin{proof}
By considering 
Lemma~\ref{lem3}, we see that it is sufficient to prove 
for any $\tau \in (0, T_{\max}),$ 
\begin{equation}\label{110}
 \|\nabla v(\ .\ ,t)\|_{L^{\infty}(\Omega)} \leq C_{2} \quad \mbox{for all} \ t \in (\tau, T_{\max}).
\end{equation}
We use the representation formula for the second equation~\eqref{yazahraadrecni}
to have:
\begin{equation*}
v(\ .\ ,t)=\textrm{e}^{t(\Delta-1)}v_{0}+\int_{0}^{t}\textrm{e}^{(t-s)(\Delta-1)}(1-u(\ .\ ,s))\, v(\ .\ ,s) \, \dd s, \quad t \in (0, T_{\max}).
\end{equation*}
We now take $p>n$ and use 
$0\leq v\leq
 \|v_{0}\|_{L^{\infty}(\Omega)}$ to write:
\begin{equation}\label{5}
\begin{aligned}
 \|(1-u(\ .\ ,s))\, v(\ .\ ,s)\|_{L^{p}(\Omega)}
 &\leq
 \|v(\ .\ ,s)\|_{L^{\infty}(\Omega)} \left(\int_{\Omega} |1-u(\ .\ ,s))|^{p} \, \dd x\right)^{\frac{1}{p}} %\nonumber
 \\
 &
 \leq
 \|v(\ .\ ,s)\|_{L^{\infty}(\Omega)} \left(\int_{\Omega} \big(1+|u(\ .\ ,s))|\,\big)^{p} \, \dd x\right)^{\frac{1}{p}} %\nonumber
 \\
 & \leq
 \|v(\ .\ ,s)\|_{L^{\infty}(\Omega)} \left(2^{p-1}
 \int_{\Omega} \big(1+|u(\ .\ ,s))|^{p}\,\big) \, \dd x\right)^{\frac{1}{p}}
% \nonumber
\\
 & \leq
 2^{\frac{p-1}{p}} \|v(\ .\ ,s)\|_{L^{\infty}(\Omega)} \left(|\Omega|^{\frac{1}{p}}
 +\|u(\ .\ ,s))\|_{L^{p}(\Omega)}\,\right) 
% \nonumber
\\
 & \leq c,
\end{aligned}
\end{equation}
where we have used the inequality $(a + b)^{m} \leq 2^{m-1}(a^{m}+b^{m})$ with $a, b \geq 0$ and $m > 1,$ also $(a + b)^{m'} \leq (a^{m'}+b^{m'})$
with $0<m'<1.$
We note that the constant $c$ in the last estimate,
it also
depends on $C_{1}$ and $
 \|v_{0}\|_{L^{\infty}(\Omega)}.$ 
In order to prove~\eqref{110},
we take $ \tau\in (0, \min \{1,T_{\max}\})$ and $\theta\in (\frac{p+n}{2\,p}, 1)$ 
and use the estimates $(3.16)$ and $(3.17)$ in~\cite{tao},
also~\eqref{5} to obtain:
\begin{align*}\label{328}
\|v(\ .\ ,t)\|_{W^{1,\infty}(\Omega)}
&\leq
c\, \|(-\Delta+1)^{\theta}v(\ .\ ,t)\|_{L^{p}(\Omega)}\nonumber\\
&\leq
c \,t^{-\theta}\textrm{e}^{-\delta t}\,\|v_{0}\|_{L^{p}(\Omega)}+
c\int_{0}^{t}(t-s)^{-\theta}\textrm{e}^{-\delta\, (t-s)}\|(1-u(\ .\ ,s))\, v(\ .\ ,s)\|_{L^{p}(\Omega)}\, \dd s\nonumber\\
&\leq 
c \, t^{-\theta}+c\int_{0}^{t}(t-s)^{-\theta}\textrm{e}^{-\delta\,(t-s)} \, \dd s\nonumber\\
&\leq
c\, t^{-\theta}+c \int_{0}^{+\infty}\sigma^{-\theta}\textrm{e}^{-\delta\, \sigma} \, \dd \sigma \nonumber\\
&\leq 
c\, (\tau^{-\theta}+1), \quad t \in (\tau, T_{\max}),
\end{align*}
where the constants can vary from line to line.
In this estimate, from the third line on-wards, the constants are also dependent on $C_{1}$
 and $
 \|v_{0}\|_{L^{\infty}(\Omega)}.$ 
This completes our proof.
\end{proof}
\end{lemma}
The proof of the following lemma is the same as~\cite[Lemma~2.5]{KHB} and similar to~\cite[Lemma~3.2]{tao}.
 But, we write it to complement our content.
 \begin{lemma}\label{lem4}
Assume that the initial values $u_{0}$
 and $ v_{0} $ are non-negative and
 satisfy: 
 \begin{equation*}
 (u_{0}, v_{0} ) \in (W^{1,r}(\Omega))^{2} 
 \quad \mbox{for some} \quad r > n.
 \end{equation*}
Also, assume that~\eqref{313*****} holds.
Then there exists some positive constant $c$ 
 such that 
for all $t \in (0, T_{\max}),$ the following
estimate holds
\begin{equation*}
\|u(\ .\ ,t)\|_{L^{\infty}(\Omega)}\leq c.
\end{equation*}
\begin{proof}
 We take $q\geq 2$ and use from~\eqref{yazahraadrecni} and integration by parts to obtain:
\begin{equation}\label{313***} 
\begin{aligned}
\frac{\dd}{\dd t}\int_{\Omega} u^{q} \, \dd x
&= q\int_{\Omega} u^{q-1} \left[\nabla\cdot( \gamma(v)\, \nabla u-u\, \xi(v)\,\nabla v)+ \mu\, u \,(1- u) \right]\, \dd x 
%\nonumber
\\
&
=-q\,(q-1)\int_{\Omega}\gamma(v)\, u^{q-2} | \nabla u|^{2} \, \dd x + q\,(q-1) \int_{\Omega} u^{q-1} \xi(v)\, \big(\nabla u\cdot \nabla v\big) \, \dd x%\nonumber
\\
& \hspace*{75mm}%\quad
+ \mu\, q \,\int_{\Omega} u^{q} \,(1- u) \, \dd x.
\end{aligned}
\end{equation}
Because of $0\leq v\leq
 \|v_{0}\|_{L^{\infty}(\Omega)}, $
we have:
\begin{align*}
 & \gamma(v)=(1+v)^{-k}\geq (1+ \|v_{0}\|_{L^{\infty}(\Omega)})^{-k}\coloneqq c_{1},\\
 & \xi(v)=k\,(1-\alpha)(1+v)^{-k-1}\leq k(1-\alpha)\coloneqq c_{2}.
\end{align*}
Making use of these,~\eqref{313*****} and Young's inequality, we can write~\eqref{313***} as follows:
\begin{equation}\label{ya mahdi}
\begin{aligned}
\frac{\dd}{\dd t}\int_{\Omega} u^{q} \, \dd x 
&\leq- c_{1}\,q\,(q-1)\int_{\Omega} u^{q-2} | \nabla u|^{2} \, \dd x + C_{2} \,c_{2} \,q\,(q-1)\int_{\Omega} u^{q-1} |\nabla u| \, \dd x 
%\nonumber
%\\
%&\hspace*{70mm} %\qquad
+\mu\, q\,\int_{\Omega} u^{q}\, \dd x
%\nonumber
\\
&%\quad 
= 
-\frac{4 \,c_{1}\,(q-1)}{q}\int_{\Omega}\big|\nabla u^{\frac{q}{2}}\big|^{2} \, \dd x +2\, C_{2} \,c_{2} \,(q-1)\int_{\Omega} u^{\frac{q}{2}}\cdot \big|\nabla u^{\frac{q}{2}}\big| \, \dd x 
%\nonumber
%\\
%&\hspace*{70mm}
+\mu\, q\,\int_{\Omega} u^{q}\, \dd x
%\nonumber
\\
&%\quad 
\leq-\frac{2\,c_{1}\,(q-1)}{q}\int_{\Omega}\big|\nabla u^{\frac{q}{2}}\big|^{2} \, \dd x+q\,\left(\frac{(q-1)\,(C_{2} \, c_{2}) ^{2}}{2\,c_{1}}+\mu \right) \int_{\Omega} u^{q}\, \dd x.
\end{aligned}
\end{equation}
We now add $q \int_{\Omega} u^{q}\, \dd x$ on both sides of~\eqref{ya mahdi} to have:
\begin{equation}\label{main essential}
\frac{\dd}{\dd t}\int_{\Omega} u^{q} \, \dd x + q \int_{\Omega} u^{q}\, \dd x \leq 
-\frac{2\,c_{1}\,(q-1)}{q}\int_{\Omega}|\nabla u^{\frac{q}{2}}|^{2} \, \dd x + c_{3}\int_{\Omega} u^{q}\, \dd x
\end{equation}
 with 
 \begin{equation*}
 c_{3}=q\,\left(\frac{(q-1)\,(C_{2} \, c_{2}) ^{2}}{2\,c_{1}}+\mu +1\right).
 \end{equation*}
We first apply the Gagliardo--Nirenberg inequality and then
use the Young inequality with exponents $s=\frac{n+2}{2}$ and $s'=\frac{n+2}{n}$
 to obtain:
\begin{align*}
c_{3}\int_{\Omega} u^{q}\, \dd x
=c_{3} \, \big\|u^{\frac{q}{2}}\big\|_{L^{2}(\Omega)}^{2}
&\leq 
c_{3} \,(C_{GN})^{2}\left(\big\|\nabla u^{\frac{q}{2}}\big\|_{L^{2}(\Omega)}^{\frac{n}{n+2}}\big\| u^{\frac{q}{2}}\big\|_{L^{1}(\Omega)}^{\frac{2}{n+2}}+\big\| u^{\frac{q}{2}}\big\|_{L^{1}(\Omega)}\right)^{2}\\
&\leq 
2\, c_{3} \,(C_{GN})^{2}\left(\big\|\nabla u^{\frac{q}{2}}\big\|_{L^{2}(\Omega)}^{\frac{2n}{n+2}}\big\| u^{\frac{q}{2}}\big\|_{L^{1}(\Omega)}^{\frac{4}{n+2}}+\big\| u^{\frac{q}{2}}\big\|_{L^{1}(\Omega)}^{2}\right)\\
&\leq
\frac{2\,c_{1}\,(q-1)}{q}\big\|\nabla u^{\frac{q}{2}}\big\|_{L^{2}(\Omega)}^{2}+\big(c_{4}+2\, c_{3}\, (C_{GN})^{2}\big)\big\| u^{\frac{q}{2}}\big\|_{L^{1}(\Omega)}^{2}\\
&
=\frac{2\,c_{1}(q-1)}{q}\int_{\Omega}|\nabla u^{\frac{q}{2}}|^{2}\, \dd x +c_{5}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{2}
\end{align*}
with
\begin{align*}
 c_{4}=\frac{1}{s}\left(\frac{2\,c_{1}\, s'\, (q-1)}{q}\right)^{-\frac{s}{s'}}\big(2\, c_{3} \,(C_{GN})^{2}\big)^{s}
 \quad \mbox{and}\quad c_{5}=c_{4}+2\, c_{3}\, (C_{GN})^{2},
 \end{align*}
 where $C_{GN}$ is the constant in the Gagliardo–Nirenberg inequality.
 Combining the last inequality with~\eqref{main essential}
 yields:
\begin{equation*}
\frac{\dd}{\dd t}\int_{\Omega} u^{q} \, \dd x +q \int_{\Omega} u^{q}\, \dd x \leq 
 c_{5} \left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{2}.
\end{equation*}
For $0 \leq t \leq T_{\max}, $ we can write:
\begin{equation*}
\frac{\dd}{\dd t}\left(\expe^{q\,t}\int_{\Omega} u^{q} \, \dd x \right) \leq c_{5}\, \expe^{q\,t}\,
 \left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{2}.
\end{equation*}
Now, we integrate and use 
$\expe^{-q\,t}\leq 1$
to get:
\begin{align*}
\int_{\Omega} u^{q}\, \dd x &
\leq \int_{\Omega} u^{q}_{0} \, \dd x+
\frac{c_{5}}{q} \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{2}\nonumber\\
 &
\leq |\Omega|\, \| u_{0}\|_{L^{\infty}(\Omega)}^{q}+
\frac{c_{5}}{q} \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{2}.
\end{align*}
Thus,
\begin{align}\label{main essential11}
\left(\int_{\Omega} u^{q} \, \dd x\right)^{\frac{1}{q}} 
 &
\leq \left[\,|\Omega|\, \| u_{0}\|_{L^{\infty}(\Omega)}^{q}+
 \frac{c_{5}}{q} \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{2}\right]^{\frac{1}{q}}\nonumber\\
 &
\leq |\Omega|^{\frac{1}{q}}\, \| u_{0}\|_{L^{\infty}(\Omega)}+
\big(\frac{c_{5}}{q}\big)^{\frac{1}{q}} \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{\frac{2}{q}}.
\end{align}
We note that
\begin{align*}
 c_{5}&=c_{4}+2\, c_{3} \,(C_{GN})^{2} \\
 &=\frac{1}{s}\left(\frac{2\,c_{1}\, s'\, (q-1)}{q}\right)^{-\frac{s}{s'}}\big(2\, c_{3} \,(C_{GN})^{2}\big)^{s}+2\, c_{3} \, (C_{GN})^{2}\\
 &=\frac{1}{s}\big(2\,c_{1}\, s'\big)^{-\frac{s}{s'}}\big(2
 \,(C_{GN})^{2}\big)^{s}
 \big(\frac{q}{q-1}\big)^{\frac{n}{2}}(c_{3})^{s}+2\, c_{3}\, (C_{G N})^{2}\\
 & \leq m \big[\big(\frac{q}{q-1}\big)^{\frac{n}{2}}(c_{3})^{s}+c_{3} \big]\\
 & \leq m \big[\big(\frac{q}{q-1}\big)^{\frac{n}{2}}+1 \big](c_{3})^{s}
\end{align*}
with
 \begin{equation*}
 m=\max\Big\{\frac{1}{s}\,\big(2\,c_{1}\, s' \,\big)^{-\frac{s}{s'}}\big(2
 \,(C_{GN})^{2}\big)^{s},2\, (C_{GN})^{2}\Big\}.
 \end{equation*}
 Here, we have used from $c_{3}>1$ and $s>1.$
 By inserting $c_{3}$ and using $q \geq 2,$
 we obtain:
\begin{equation}\label{salam}
\begin{aligned}
 \frac{c_{5}}{q}&\leq m \left[\big(\frac{q}{q-1}\big)^{\frac{n}{2}}+1 \right]\left(\frac{(q-1)\,(C_{2}\,c_{2}) ^{2}}{2\,c_{1}}+\mu +1\right)^{\frac{n}{2}+1} q^{\frac{n}{2}}%\nonumber
 \\
 & \leq 
 2\, m \,\left(\frac{ (C_{2}\,c_{2})^{2}}{2\,c_{1}}+\mu +1\right)^{\frac{n}{2}+1}\, \big(\frac{q}{q-1}\big)^{\frac{n}{2}} (q-1)^{\frac{n}{2}+1}\, q^{\frac{n}{2}}%\nonumber
 \\
 &= c_{6} \, (q-1)\, q^{n} %\nonumber
 \\
 & \leq c_{6} \, q^{n+1}
\end{aligned}
\end{equation}
with
\[
 c_{6}= 2\, m \,\left(\frac{ (C_{2}\,c_{2}) ^{2}}{2\,c_{1}}+\mu +1\right)^{\frac{n}{2}+1}. 
\]
Making use of~\eqref{salam} and $q^{\frac{n+1}{q}}>1,$ we can write 
\eqref{main essential11} as follows:
\begin{equation}\label{yalatif}
\begin{aligned}
\left(\int_{\Omega} u^{q} \, \dd x\right)^{\frac{1}{q}} 
 &
\leq |\Omega|^{\frac{1}{q}}\, \| u_{0}\|_{L^{\infty}(\Omega)}+
\big( c_{6} \, q^{n+1}\big)^{\frac{1}{q}} \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{\frac{2}{q}}
%\nonumber
\\
&
\leq c_{7}^{\frac{1}{q}} \, q^{\frac{n+1}{q}} \left( \| u_{0}\|_{L^{\infty}(\Omega)}+
 \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{\frac{q}{2}}\, \dd x\right)^{\frac{2}{q}}\right)
\end{aligned}
\end{equation}
with $c_{7}=|\Omega|+ c_{6}.$
We now define:
\begin{equation*}
M(q)=\max\left\lbrace \| u_{0}\|_{L^{\infty}(\Omega)}, \sup_{0\leq t\leq T_{\max}}\left(\int_{\Omega} u^{q}\, \dd x\right)^{\frac{1}{q}}\right\rbrace. 
\end{equation*}
This allows us to write~\eqref{yalatif} as:
\begin{equation*}
M(q)\leq 2\, c_{7}^{\frac{1}{q}} \, q^{\frac{n+1}{q}} M\left(\frac{q}{2}\right).
\end{equation*}
We now take $q= 2^{i}$
$(i\in\mathbb{N})$ to obtain:
\begin{equation}\label{yakarim}
\begin{aligned}
M(2^{i}) & \leq 2\, c_{7}^{2^{-i}}
 \, 2^{\frac{(n+1) i}{2^{i}}} M(2^{i-1}) %\nonumber
 \\
& \leq 2\, c_{7}^{2^{-i}+2^{-i+1}}
 2^{(n+1)\big(\frac{i}{2^{i}}+\frac{i-1}{2^{i-1}}\big)} 
M(2^{i-2}) %\nonumber
\\
& \leq \cdots %\nonumber 
\\
& \leq 2\, c_{7}^{2^{-i}+2^{-i+1}+\cdot+2^{-1}}
 2^{(n+1)\big(\frac{i}{2^{i}}+\frac{i-1}{2^{i-1} }+\cdot+\frac{1}{2}\big)} 
M(1),
\end{aligned}
\end{equation} 
and compute the following elementary series:
\begin{align*}
 S\coloneqq \sum_{i=1}^{\infty}\frac{i}{2^{i}}&= \sum_{i=0}^{\infty}\frac{i+1}{2^{i+1}}=\sum_{i=0}^{\infty}\big(\frac{i}{2^{i+1}}+\frac{1}{2^{i+1}}\big)=\frac{1}{2}\sum_{i=1}^{\infty}\frac{i}{2^{i}}+\sum_{i=0}^{\infty}\frac{1}{2^{i+1}}=\frac{1}{2}S+1.
\end{align*}
Thus, $S=2.$ Making use of this, $\lim_{i \rightarrow \infty}\| u(\ .\ ,t)\|_{L^{2^{i}}(\Omega)}=\|u(\ .\ ,t)\|_{L^{\infty}(\Omega)} $
and~\eqref{mamnonam}, by letting $i \rightarrow \infty$ in~\eqref{yakarim},
we obtain the desired result.
\end{proof}
\end{lemma}
We now can write our main theorem.
\begin{theo}	\label{thm}
 Let the initial values $u_{0}$
 and $ v_{0} $ are non-negative and
 satisfy: 
 \begin{equation*}
 (u_{0}, v_{0} ) \in (W^{1,r}(\Omega))^{2} 
 \quad \mbox{for some} \quad r > n
 \end{equation*}
 and 
 \begin{equation*}
 0<\|v_{0}\|_{L^{\infty}(\Omega)}\leq \delta_{0},
 \end{equation*}
 which $\delta_{0}$ is introduced in Lemma~\ref{lemk}.
 Assume that the condition~\eqref{sallamkhoda} holds. 
Then, 
 the solution of the problem~\eqref{yazahraadrecni} with the functions $\gamma(v)$
and $\xi(v)$
	 defined by~\eqref{yarahim+}
 is global and bounded.
\begin{proof}
 By considering the extensibility criterion provided by Lemma~\ref{lem3}, the proof is a consequence of~\eqref{yaallah} and Lemma~\ref{lem4}.\\
 \end{proof}
\end{theo}


\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no
affiliations other than their research organizations.

\printbibliography


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