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\TopicEN{Partial differential equations}

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\title{Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source}

\author{\firstname{Khadijeh} \lastname{Baghaei}}
%\address{Iran, Islamic Republic of}
\address{Pasargad Institute for Advanced Innovative Solutions, No.30, Hakim Azam St., North Shiraz St., Mollasadra Ave., Tehran, Iran}
\email{kh.baghaei@gmail.com}

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

\begin{abstract}
We consider the 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
\text{and} \quad
\xi(v)=-(1-\alpha)\gamma'(v),
\end{equation*}
where $k>0$ and $\alpha \in (0,1).$

We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $ k\,(1-\alpha)<\frac{4}{n+5}$ and the initial value $ v_{0}$ and $\mu$ satisfy the following conditions:
\begin{align*}
0<\|v_{0}\|_{L^{\infty}(\Omega)}\leq \Bigg[\frac{4\big[1-k \, \big(1-\alpha\big)\big]}{k\, (n+1)\,(1-\alpha)}\Bigg]^{\frac{1}{k}}-1,
\end{align*}
and
\begin{equation*}
\mu>
\frac{k\,n\,(1-\alpha) \|v_{0}\|_{L^{\infty}(\Omega)}}{(n+1)(1+\|v_{0}\|_{L^{\infty}(\Omega)})}.
\end{equation*}
This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023).
\end{abstract}


\begin{document}
\maketitle

\section{Introduction}

In this paper, we study 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. Here, $u=u(x, t)$ denotes the cell density and $v=v(x, t)$ is the nutrient consumed chemical concentrations.

In mathematical biology, systems like~\eqref{yazahraadrecni} describe the mechanism of chemotaxis. The chemotaxis is the movement of cells towards a higher concentration of a chemical signal substance produced by the cells. If the second equation of problem~\eqref{yazahraadrecni} is changed and 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 $\tau \in \{0,1\},$ then this system is the classical chemotaxis system which has been introduced by Keller and Segel~\cite{ks}. For problem~\eqref{145}, in the absence of logistic source, when the positive function $\gamma(v)$ belongs to $ C^{3}((0, \infty))$ and $ \xi(v)=-\gamma'(v)$ as well as
\begin{align*}
\gamma_{\infty}:=\limsup _{s \rightarrow \infty} \gamma(s) < \frac{1}{\tau},
\end{align*}
then for $n \geq 1,$ the existence a unique global non-negative classical solution is proved~\cite{24}. Also, the uniform-in-time boundedness of classical solutions is proved in any dimension when the function $\gamma$ has strictly positive lower and upper bounds~\cite{24}. This result also is proved for $n \geq 2,$ when the function $\gamma$ decays at a certain slow rate at infinity~\cite{24}.

In the special case $\gamma(v)= c_{0} \, v^{-k}$ with $k>0$ and $c_{0} >0,$ for $n \geq 1,$ the global existence and boundedness of the solution is proved for all $k>0$ under a smallness assumption on $c_{0}$~\cite{Yk}. When $n \geq 2,$ by removing the smallness condition on $c_{0},$ and applying the condition $k \in (0, \frac{2}{n-2}),$ the same result is proved in cases $\tau=0$~\cite{AY} and $\tau=1$~\cite{FS}.

In the other special case $\gamma(v)= \eit^{-\chi v}$ with $ \chi>0,$ for $n=2,$ it is proved that the classical solutions for this problem are global and bounded if $\int_{\Omega} u_{0} \, \dx < \frac{4 \pi}{\chi},$ whereas for $\int_{\Omega} u_{0} \, \dx > \frac{4 \pi}{\chi}$ blow up occurs either in finite or infinite time~\cite{2020}. For $n=2$ and $\tau=0,$ it is proved that the blow up occurs in infinite time~\cite{Fj}. Also, for $n=2,$ it is proved that the classical solution is globally bounded if the positive function $\gamma(v)$ decreases slower than an exponential speed at high signal concentrations. For $n \geq 3,$ this result is proved when $\gamma(v)$ decreases at certain algebraically speed~\cite{2021}. Also, in the presence of logistic source, when $n=2$ and the positive function $\gamma(v)$ belongs to $ C^{3}([0, \infty)),$ $\gamma'(v)<0,$ $\lim _{v \rightarrow \infty}\gamma(v)=0 $ and $\lim _{v \rightarrow \infty}\frac{\gamma'(v)}{\gamma(v)} $ exists, the existence of bounded classical solutions are proved in~\cite{55}. For $n \geq 3,$ if the last condition is replaced with $|\gamma'(v)| \leq m,$ where $m$ is some positive constant, then the global existence and boundedness of the solution is proved when $\mu >0$ is large~\cite{2019}.

Now, we want to write some results related to problem~\eqref{yazahraadrecni}. But first, we explain the origin of the definition of this problem. Tuval et\,al. in~\cite{2005} introduced 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. We see that problem~\eqref{yazahraadrecni} can be obtained from the preceding chemotaxis-Navier--Stokes system upon the choice $\omega\equiv 0, \gamma(v)\equiv 1$ and $ g(v)=v.$ For the related results with the chemotaxis-Navier--Stokes systems, we refer the interested readers to~\cite{2021w, 2014w,2016l, 2023} and references therein. 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, in the two-dimensional case for the bounded convex domains with smooth boundary, it is proved that the classical solutions for this problem are global and bounded~\cite{taowin}. Also, for $n \geq 3,$ the classical solutions for this problem are global and bounded provided that $\|v_{0}\|_{L^{\infty}(\Omega)}\leq \frac{1}{6\,(n+1)\chi}$~\cite{tao}. This condition extends to $\|v_{0}\|_{L^{\infty}(\Omega)}\leq \frac{\pi}{\chi\, \sqrt{2(n+1)}}$ in~\cite{baghaei-khelghati}. Also, in the presence of logistic source under this condition, the existence of bounded classical solutions is proved in~\cite{baghaei-khelghati-logistic}.

The authors in~\cite{5} studied the problem~\eqref{yazahraadrecni} when the positive function $\gamma(v)$ belongs to $ C^{3}([0, \infty))$ and $\gamma'(v)<0$ for all $v \geq 0$ as well as $ \xi(v)=-\gamma'(v).$ For $n = 2$ and $\mu >0,$ they proved the global existence and boundedness of solution. Also, when $n \geq 3$ and $\mu$ is suitably large, they obtained the same result. Besides, they showed the solution converges exponentially to $(1, 0)$ when $t$ tends to infinity. In the case of $\mu=0,$ under the same conditions on $\gamma(v),$ the authors in~\cite{6} proved the existence a unique global bounded classical solution with some suitable small initial data. Wang in~\cite{7} studied the above problem when the logistic source is as $f(u)=\alpha\, u -\mu \,u^{\kappa}.$ He proved that this problem admits a global bounded classical solution 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.

In~\cite{4}, the authors studied the problem~\eqref{yazahraadrecni}. They assumed that the positive function $\gamma(v)$ belongs to $C^{2}([0, \infty))$ such that $\gamma'(v)<0,$ $\gamma''(v)\geq 0$ and $ \xi(v)=-(1-\alpha)\gamma'(v)$ with $\alpha \in (0,1).$ Under the following conditions:
\begin{align*}
& \frac{(\gamma'(v))^{2}}{\gamma''(v)}
\leq \frac{n}{2(n+1)^{3}}, \qquad 0< \|v_{0}\|_{L^{\infty}(\Omega)}\leq \gamma^{-1}\Biggl(\frac{1}{n+1}\Biggr)
\end{align*}
and
\begin{align*}
\mu > \max_{0< v \leq \|v_{0}\|_{L^{\infty}(\Omega)}}
\frac{-\gamma'(v)\, \|v_{0}\|_{L^{\infty}(\Omega)}}{\gamma(v)},
\end{align*}
they proved that the problem~\eqref{yazahraadrecni} has a unique global classical solution that is uniformly in time bounded. Besides, under some conditions, they proved that the solution converges to $(1, 0)$ when $t$ tends to infinity. For this problem, there are other results. To see these results, we refer the interested readers to~\cite{taowin2, 8, 11} and references therein. In this paper, we focus on the functions $\gamma(v)$ and $\xi(v)$ as follows:
\begin{equation}\label{yarahim+}
\gamma(v)=(1+v)^{-k}\quad
\text{and} \quad
\xi(v)=-(1-\alpha)\gamma'(v)
\end{equation}
where $k>0$ and $\alpha \in (0,1).$ For these functions, we will prove the following theorem:

\begin{theo}\label{thm}
Let $u_{0} \geq 0$ and $v_{0}\geq0$ satisfy $(u_{0}, v_{0}) \in (W^{1,q}(\Omega))^{2}$ for some $q > n$ and the functions $\gamma(v)$ and $\xi(v)$ are defined as~\eqref{yarahim+}. Also, assume that
\begin{align}\label{3131}
k\,(1-\alpha)<\frac{4}{n+5}
\end{align}
and the initial value $ v_{0}$ and $\mu$ satisfy the following conditions:
\begin{align}\label{sallam}
0<\|v_{0}\|_{L^{\infty}(\Omega)}\leq \Bigg[\frac{4\big[1-k \, \big(1-\alpha\big)\big]}{k\, (n+1)\,(1-\alpha)}\Bigg]^{\frac{1}{k}}-1
\end{align}
and
\begin{equation}\label{sallamkhoda}
\mu>
\frac{k\,n\,(1-\alpha) \|v_{0}\|_{L^{\infty}(\Omega)}}{(n+1)(1+\|v_{0}\|_{L^{\infty}(\Omega)})}.
\end{equation}
Then the solution of the problem~\eqref{yazahraadrecni} is global and bounded.
\end{theo}

We note that the authors in~\cite{4} in the case of $\gamma(v)=(1+v)^{-k}(k>0)$ proved that the solution of the problem~\eqref{yazahraadrecni} is global and bounded provided that:
\begin{align*}
k<\frac{n}{2(n+1)^{3}-n},
\quad \quad
\mu>k\|v_{0}\|_{L^{\infty}(\Omega)}
\qquad \text{and} \qquad
\|v_{0}\|_{L^{\infty}(\Omega)}\leq \gamma^{-1}\Biggl(\frac{1}{n+1}\Biggr).
\end{align*}
Because of $\gamma'(v)<0,$ the last condition is written as:
\begin{align*}
0< \|v_{0}\|_{L^{\infty}(\Omega)}\leq (n+1)^{\frac{1}{k}}-1.
\end{align*}
In the following, we show that our result improves the obtained result in~\cite{4}.
\begin{itemize}
\item For $\alpha \in (0,1),$ it is not difficult to see that:
\begin{align*}
\frac{n}{2(n+1)^{3}-n}<\frac{4}{n+5} <\frac{4}{(n+5)(1-\alpha)}.
\end{align*}
Thus, our result extends the range of $k.$
\item We see that if $k<\frac{4}{((n+1)^{2}+4)(1-\alpha)},$ then:
\begin{align*}
n+1< \frac{4\big[1-k \, \big(1-\alpha\big)\big]}{k\, (n+1)\,(1-\alpha)}.
\end{align*}
Because of
\begin{align*}
\frac{n}{2(n+1)^{3}-n}<\frac{4}{(n+1)^{2}+4} <\frac{4}{((n+1)^{2}+4)(1-\alpha)},
\end{align*}
therefore, our result extends the upper bound obtained for $\|v_{0}\|_{L^{\infty}(\Omega)}$ corresponding to the range of $k$ in~\cite{4}.
\item Also, we have
\begin{align*}
\frac{n\,(1-\alpha)}{(n+1)(1+\|v_{0}\|_{L^{\infty}(\Omega)})}<1.
\end{align*}
Thus, if we take the values $k$ and $\|v_{0}\|_{L^{\infty}(\Omega)}$ in the range of obtained in~\cite{4}, then the lower bound obtained in our result for $\mu$ is smaller than the lower bound obtained in~\cite{4}.
\end{itemize}

\section{Our results}

Here, we state the standard well-posedness and classical solvability result.

\begin{lemm}\label{lem3}
Let $u_{0} \geq 0$ and $v_{0}\geq 0$ satisfy $(u_{0}, v_{0}) \in (W^{1,q}(\Omega))^{2}$ for some $q > n.$ Then problem~\eqref{yazahraadrecni} has a unique local in time classical solution
\begin{equation*}
(u,v) \in \Big(C \big([0, T_{max}); W^{1,q}(\Omega)\big) \cap C^{2,1}\big(\,\overline{\Omega}\times (0, T_{max})\Big)^{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(\,\cdot\,,t)\|_{L^{\infty}(\Omega)}= + \infty.
\end{equation*}
Moreover, $u$ and $v$ satisfy the following inequalities:
\begin{equation}\label{yaallah1}
u \geq 0 \quad \text{and} \quad 0 \leq v \leq \|v_{0}\|_{L^{\infty}(\Omega)}
\quad \text{in} \quad \Omega\times (0, T_{max}),
\end{equation}
also,
\begin{align}\label{mamnonam}
\int_{\Omega} u(\,\cdot\,, t) \, \dx \leq c,
\end{align}
where $c$ is some positive constant.
\end{lemm}

For details of the proof, we refer the reader to~\cite{55, 4}.

Based on main idea in~\cite{baghaei-khelghati, baghaei-khelghati-logistic, KB1, KB2}, we write the following key lemma.

\begin{lemm}\label{lem1}
Let $(u, v)$ be the solution of problem~\eqref{yazahraadrecni}. If there exists a smooth positive function $\varphi(v)$ such that for $p \geq 2$ the following inequality holds
\begin{equation}\label{yaallah}
(B(v))^{2}-4 A(v) C(v) \leq 0,
\end{equation}
where the functions $A, B$ and $C$ are defined as:
\begin{equation}\label{313}
\left\{
\begin{aligned}
A(v)&=(p-1) \varphi (v) \gamma(v), \\
B(v)&=(p-1) \varphi (v) \xi(v) -\varphi' (v) (\gamma(v)+1), \\
C(v)&= \tfrac{1}{p} \varphi ''(v)- \varphi' (v) \xi(v),
\end{aligned}
\right.
\end{equation}
then:
\begin{equation*}
\frac{1}{p}\frac{\dd}{\dt}\int _{\Omega} u^{p} \varphi (v) \, \dx
\leq -\int _{\Omega} \Biggl[\mu \varphi (v)+\frac{1}{p} v \varphi' (v)\Biggr] u^{p+1}\, \dx +\mu \int_{\Omega}u^{p} \varphi (v)\, \dx.
\end{equation*}
\end{lemm}

\begin{proof}
We assume that there exists a smooth positive function $\varphi(v)$ such that for $p\geq 2,$~\eqref{yaallah} holds. We take this function and use~\eqref{yazahraadrecni} and integration by parts to write:
\begin{align}\label{yasahebazzaman}
\frac{1}{p}\frac{\dd}{\dt}\int _{\Omega} u^{p} \varphi (v) \, \dx & =
\int _{\Omega}u^{p-1} \varphi (v)\ u_{t} \ \dx + \frac{1}{p}\int_{\Omega}u^{p} \ \varphi' (v) v_{t} \, \dx \nonumber\\
&= - (p-1) \int_{\Omega}u^{p-2} \varphi (v) \gamma(v) \, |\nabla u|^{2} \ \dx\nonumber\\
& \qquad + \int _{\Omega}u^{p-1} \Big[(p-1) \varphi (v) \xi(v) -\varphi' (v) (\gamma(v)+1) \Big]
\big(\nabla u \cdot \nabla v\big) \, \dx\nonumber\\
& \qquad + \int _{\Omega}u^{p} \Biggl[\varphi' (v) \xi(v) - \frac{1}{p} \varphi ''(v) \Biggr] |\nabla v |^{2} \, \dx \nonumber\\
& \qquad - \int _{\Omega} \Biggl[\mu \varphi (v)+\frac{1}{p} v \varphi' (v)\Biggr] u^{p+1}\, \dx +\mu \int_{\Omega}u^{p} \varphi (v)\, \dx.
\end{align}
For convenience in calculations, we write~\eqref{yasahebazzaman} as follows:
\begin{equation}\label{yarahim}
\frac{1}{p}\frac{\dd}{\dt}\int _{\Omega} u^{p} \varphi (v) \, \dx = \int _{\Omega} J (u, v) \, \dx - \int _{\Omega} \Biggl[\mu \varphi (v)+\frac{1}{p} v \varphi' (v)\Biggr] u^{p+1}\, \dx +\mu \int_{\Omega}u^{p} \varphi (v)\, \dx
\end{equation}
with
\begin{align}\label{ya mahdi adrecni}
J(u, v)&= - (p-1) u^{p-2} \varphi (v) \gamma(v) \, |\nabla u|^{2} \nonumber\\
& \qquad +u^{p-1} \Big[(p-1) \varphi (v) \xi(v) -\varphi' (v) (\gamma(v)+1) \Big]
\big(\nabla u \cdot \nabla v\big) \nonumber\\
& \qquad + u^{p} \Biggl[\varphi' (v) \xi(v) - \frac{1}{p} \varphi ''(v) \Biggr] |\nabla v |^{2} \nonumber\\
&= - u^{p-2} A(v) |\nabla u |^{2} +u^{p-1} B(v)
\big(\nabla u \cdot
\nabla v \big) - u^{p} C(v)|\nabla v |^{2},
\end{align}
where $ A, B $ and $ C $ are defined as~\eqref{313}. Now, by considering~\eqref{ya mahdi adrecni}, we can write
\begin{align*}
J(u,v) & =-\Biggl(\sqrt{ u^{p-2} A(v)} \nabla u - \frac{u^{p-1}B(v)}{2\sqrt{ u^{p-2} A(v)}} \nabla v\Biggr)\cdot\Biggl(\sqrt{ u^{p-2} A(v)} \nabla u -\frac{u^{p-1}B(v)}{2\sqrt{ u^{p-2} A(v)}} \nabla v\Biggr)
\\
&\qquad +u^{p}\Biggl[\frac{(B(v))^{2}}{4 A(v)}-C(v)\Biggr]|\nabla v |^{2}\quad \
\\
&\leq u^{p}\Biggl[\frac{(B(v))^{2}-4A(v) C(v)}{4 A(v)} \Biggr]|\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}{\dt}\int _{\Omega} u^{p} \varphi (v) \, \dx
\leq -\int _{\Omega} \Biggl[\mu \varphi (v)+\frac{1}{p} v \varphi' (v)\Biggr] u^{p+1}\, \dx +
\mu \int_{\Omega}u^{p} \varphi (v)\, \dx.
\end{equation*}
This completes our proof.
\end{proof}

In the following lemma, we present a function $\varphi$ and show that for this function, the relation~\eqref{yaallah} holds.

\begin{lemm}\label{main}
Let $u_{0} \geq 0$ and $v_{0}\geq0$ satisfy $(u_{0}, v_{0}) \in (W^{1,q}(\Omega))^{2}$ for some $q > n$ and the functions $\gamma(v)$ and $\xi(v)$ are defined as~\eqref{yarahim+}. Also, assume that~\eqref{3131}, \eqref{sallam} and~\eqref{sallamkhoda} hold. Then there exists some positive constant $c$ such that the first component of problem~\eqref{yazahraadrecni} for all $t \in (0, T_{\max})$ satisfies
\begin{equation}\label{313*}
\|u(\,\cdot\,,t)\|_{L^{n+1}(\Omega)}\leq c.
\end{equation}
\begin{proof}
We want to apply Lemma~\ref{lem1}. Hence, at first, we take $p=n+1$ and define the function $\varphi$ as:
\begin{equation*}
\varphi(v)=(1+v)^{-k\lambda}\quad \text{with}\quad \lambda=n\,(1-\alpha).
\end{equation*}
For this function, we have:
\begin{align*}
\varphi'(v)&=-k \, \lambda\, (1+v)^{-k\lambda-1}
\end{align*}
and
\begin{align*}
\varphi''(v)&=k\, \lambda\,(k\, \lambda+1)(1+v)^{-k\lambda-2}.
\end{align*}
In the following, we show that for this function $\varphi,$ the relation~\eqref{yaallah} holds. We know from~\eqref{yarahim+} that $\gamma(v)=(1+v)^{-k}$ and $\xi(v)=k\,(1-\alpha)(1+v)^{-k-1}.$ By considering these, we compute:
\begin{align*}
& (B(v))^{2}-4 A(v) C(v)\nonumber\\
&= n^{2} (\varphi(v)) ^{2} (\xi (v)) ^{2}+ (\varphi' (v)) ^{2} (\gamma(v)+1)^{2}\nonumber\\
& \quad -2\,n \,\varphi(v)\, \varphi' (v)\, \xi (v)\,(1-\gamma(v)) -\frac{4 n}{n+1} \varphi (v) \varphi ''(v) \gamma(v)\nonumber\\
&= k^{2}\,n^{2}(1-\alpha)^{2} (1+v)^{-2(k+1)-2k\,n\,(1-\alpha)}\\
& \quad + k^{2}\,n^{2}(1-\alpha)^{2} (1+v)^{-2(kn\,(1-\alpha)+1)} \big[(1+v)^{-2k}+2\,(1+v)^{-k}+1 \big]\nonumber\\
& \quad +2\, k^{2}\,n^{2}(1-\alpha)^{2} \, (1+v)^{-2kn\,(1-\alpha)-k-2} -2\, k^{2}\,n^{2}(1-\alpha)^{2} \, (1+v)^{-2kn\,(1-\alpha)-2k-2}\\
& \quad -\frac{4\, k\, n^{2}(1-\alpha)}{n+1}(k\, n\,(1-\alpha)+1)(1+v)^{-2kn\,(1-\alpha)-k-2}\\
&=k\,n^{2} (1-\alpha)(1+v)^{-2(k\, n\,(1-\alpha)+1)}\Bigg\{ k\,(1-\alpha) (1+v)^{-2k} + k\,(1-\alpha) \big[(1+v)^{-2k}+2\,(1+v)^{-k}+1 \big]\\
& \quad +2\, k\,(1-\alpha) \, \, (1+v)^{-k} -2\, k\,(1-\alpha) (1+v)^{-2k} -\frac{4}{n+1}\big(k\, n\,(1-\alpha)+1\big)(1+v)^{-k}\Bigg\}\\
&= k\,n^{2} (1-\alpha)(1+v)^{-2(k\, n\,(1-\alpha)+1)}\Bigg\{ 4\Biggl[k\,(1-\alpha) -\frac{k\, n(1-\alpha)+1}{n+1}\Biggr] (1+v)^{-k}+k\,(1-\alpha)\Bigg\}\\
&=\frac{k\,n^{2} (1-\alpha)(1+v)^{-2(k\, n\,(1-\alpha)+1)} }{n+1}
\Bigg\{k(n+1)(1-\alpha)-4[1-k\,(1-\alpha)](1+v)^{-k}
\Bigg\}
\nonumber\\
& \leq
\frac{k\,n^{2} (1-\alpha)(1+v)^{-2(k\, n\,(1-\alpha)+1)} }{n+1}
\Bigg\{k(n+1)(1-\alpha)-4[1-k\,(1-\alpha)](1+ \|v_{0}\|_{\infty})^{-k}
\Bigg\}.
\end{align*}
Under the condition~\eqref{sallam}, we see that
\begin{align*}
(B(v))^{2}-4 A(v) C(v)\leq 0.
\end{align*}
Thus, the relation~\eqref{yaallah} holds. We now can apply Lemma~\ref{lem1} and write:
\begin{align}\label{khodaya shokr}
&\frac{1}{n+1}\,\frac{\dd}{\dt}\int _{\Omega} u^{n+1} (1+v)^{-k\,n\,(1-\alpha)} \, \dx+ \mu \int_{\Omega}u^{n+1} (1+v)^{-k\,n\,(1-\alpha)}\, \dx \nonumber\\
&\quad\leq -\int _{\Omega}
\Biggl[\mu (1+v) -\frac{k\,n\,(1-\alpha)}{n+1} v \Biggr] (1+v)^{-k\,n\,(1-\alpha)-1}u^{n+2}\, \dx+ 2\mu \int_{\Omega}u^{n+1} (1+v)^{-k\,n\,(1-\alpha)}\, \dx.
\end{align}
The Young inequality allows us to write:
\begin{align}\label{shokr}
2\mu \int_{\Omega}u^{n+1} (1+v)^{-k\,n\,(1-\alpha)}\, \dx &
\leq \epsilon \int_{\Omega}u^{n+2} (1+v)^{-k\,n\,(1-\alpha)}\, \dx\, + c(\epsilon) \int_{\Omega}(1+v)^{-k\,n\,(1-\alpha)}\, \dx\nonumber \\
&
\leq \epsilon \int_{\Omega}u^{n+2} (1+v)^{-k\,n\,(1-\alpha)}\, \dx+ c\,(\epsilon) \,|\Omega|,
\end{align}
where $\epsilon$ is chosen as follows:
\begin{equation*}
0< \epsilon <\mu-
\frac{k\,n\,(1-\alpha) \|v_{0}\|_{L^{\infty}(\Omega)}}{(n+1)(1+\|v_{0}\|_{L^{\infty}(\Omega)})}
\end{equation*}
and:
\begin{equation*}
c\,(\epsilon)=\frac{1}{n+2}\Biggl[\frac{n+1}{\epsilon\,(n+2)}\Biggr]^{n+1}(2\mu)^{n+2}.
\end{equation*}

We now combine the inequality~\eqref{shokr} with~\eqref{khodaya shokr} and use from $0\leq v\leq
\|v_{0}\|_{L^{\infty}(\Omega)} $ and~\eqref{sallamkhoda} to obtain:
\begin{align*}
\frac{1}{n+1}\,\frac{\dd}{\dt}\int _{\Omega} u^{n+1} &(1+v)^{-k\,n\,(1-\alpha)} \, \dx + \mu \int_{\Omega}u^{n+1} (1+v)^{-k\,n\,(1-\alpha)}\, \dx
\nonumber\\
&\leq \int _{\Omega}
\Biggl[\epsilon-\mu+\frac{k\,n\,(1-\alpha)v}{(n+1)(1+v)} \Biggr] (1+v)^{-k\,n\,(1-\alpha)}u^{n+2}\, \dx + c(\epsilon) |\Omega|
\nonumber\\
&\leq \int _{\Omega}
\Biggl[\epsilon-\mu+\frac{k\,n\,(1-\alpha)\|v_{0}\|_{L^{\infty}(\Omega)}}{(n+1)(1+\|v_{0}\|_{L^{\infty}(\Omega)})}\Biggr] (1+v)^{-k\,n\,(1-\alpha)}u^{n+2}\, \dx + c\,(\epsilon)\, |\Omega|.
\end{align*}
We put:
\begin{equation*}
y(t)=\int _{\Omega} u^{n+1} (1+v)^{-k\,n\,(1-\alpha)} \, \dx.
\end{equation*}
We see that the value of $\epsilon$ allows us to write:
\begin{align*}
y'(t) +\mu\, (n+1)\, y(t)
\leq c\,(\epsilon)\,(n+1) |\Omega|.
\end{align*}
This yields:
\begin{align}\label{14}
y(t)
\leq \max\Biggl\{y(0), \frac{c\,(\epsilon)\, |\Omega|}{\mu}
\Biggr\}.
\end{align}
Making use of $0\leq v \leq \|v_{0}\|_{L^{\infty}(\Omega)}$ and~\eqref{14}, we have:
\begin{equation*}
\int _{\Omega} u^{n+1} \, \dx \leq \big(1+\|v_{0}\|_{L^{\infty}(\Omega)}\big)^{k\,n\,(1-\alpha)}\max\Biggl\{y(0), \frac{c\,(\epsilon)\, |\Omega|}{\mu\, }
\Biggr\}.
\end{equation*}
Thus, we obtain the desired result.
\end{proof}
\end{lemm}

The proof of the following lemma is the same as~\cite[Lemma~3.2]{tao}. But, we write it to complement our content.
\begin{lemm}
Let $u_{0} \geq 0$ and $v_{0}\geq0$ satisfy $(u_{0}, v_{0}) \in (W^{1,q}(\Omega))^{2}$ for some $q > n.$ Also, assume that~\eqref{3131}, \eqref{sallam} and~\eqref{sallamkhoda} hold. Then there exists some positive constant $C$ such that
\begin{equation}\label{313*****}
\|\nabla v\|_{L^{\infty}(\Omega)} \leq C
\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(\,\cdot\,,t)\|_{L^{\infty}(\Omega)} \leq C\quad \text{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(\,\cdot\,,t)=\erm^{t(\Delta-1)}v_{0}+\int_{0}^{t}\erm^{(t-s)(\Delta-1)}(1-u(\,\cdot\,,s)) v(\,\cdot\,,s) \, \ds, \quad t \in (0, T_{max}).
\end{equation*}
We now take $p=n+1$ and use from $0\leq v\leq
\|v_{0}\|_{L^{\infty}(\Omega)} $ and~\eqref{313*} to write:
\begin{align}\label{5}
\|(1-u(\,\cdot\,,s)) v(\,\cdot\,,s)\|_{L^{p}(\Omega)} &\leq
\|v(\,\cdot\,,s)\|_{L^{\infty}(\Omega)} \Biggl(\int_{\Omega} |1-u(\,\cdot\,,s))|^{p} \, \dx\Biggr)^{\frac{1}{p}}\nonumber\\
&
\leq
\|v(\,\cdot\,,s)\|_{L^{\infty}(\Omega)} \Biggl(\int_{\Omega} \big(1+|u(\,\cdot\,,s))|\,\big)^{p} \, \dx\Biggr)^{\frac{1}{p}}\nonumber\\
& \leq
\|v(\,\cdot\,,s)\|_{L^{\infty}(\Omega)} \Biggl(2^{p-1}
\int_{\Omega} \big(1+|u(\,\cdot\,,s))|^{p}\,\big) \, \dx\Biggr)^{\frac{1}{p}}
\nonumber\\
& \leq 2^{\frac{p-1}{p}} \|v(\,\cdot\,,s)\|_{L^{\infty}(\Omega)} \Big(|\Omega|^{\frac{1}{p}} +|\|u(\,\cdot\,,s))\|_{L^{p}(\Omega)}\,\Big)
\nonumber\\
& \leq c,
\end{align}
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 $0<m'<1.$ In order to prove~\eqref{110}, we take $ \tau\in (0, \min \{1,T_{\max}\})$ and $\theta\in \bigl(\frac{2n+1}{2(n+1)}, 1\bigr)$ and use the estimates $(3.16)$ and $(3.17)$ in~\cite{tao}, also~\eqref{5} to obtain:
\begin{align*}\label{328}
\|v(\,\cdot\,,t)\|_{W^{1,\infty}(\Omega)} &\leq c\, \|(-\Delta+1)^{\theta}v(\,\cdot\,,t)\|_{L^{p}(\Omega)}\nonumber\\
&\leq c \,t^{-\theta}\erm^{-\delta t}\,\|v_{0}\|_{L^{p}(\Omega)}+ c\int_{0}^{t}(t-s)^{-\theta}\erm^{-\delta(t-s)}\|(1-u(\,\cdot\,,s)) v(\,\cdot\,,s)\|_{L^{p}(\Omega)}\, \ds\nonumber\\
&\leq c \, t^{-\theta}+c\int_{0}^{t}(t-s)^{-\theta}\erm^{-\delta(t-s)} \ds\nonumber\\
&\leq c\, t^{-\theta}+c\int_{0}^{+\infty}\sigma^{-\theta}\erm^{-\delta \sigma} \dd\sigma \nonumber\\
&\leq c\, (\tau^{-\theta}+1), \quad t \in (\tau, T_{max}),
\end{align*}
where the constant $c$ can vary from line to line. This completes our proof.
\end{proof}
\end{lemm}

Upon the well-known Moser Alikakos iteration procedure~\cite{3}, we prove the following lemma similar to~\cite[Lemma~3.2]{tao}.

\begin{lemm}\label{lem4}
Let $u_{0} \geq 0$ and $v_{0}\geq0$ satisfy $(u_{0}, v_{0}) \in (W^{1,q}(\Omega))^{2}$ for some $q > n.$ Also, assume that~\eqref{3131}, \eqref{sallam} and~\eqref{sallamkhoda} hold. Then there exists some positive constant $c$ such that the first component of problem~\eqref{yazahraadrecni} for all $t \in (0, T_{\max})$ satisfies
\begin{equation*}
\|u(\,\cdot\,,t)\|_{L^{\infty}(\Omega)}\leq c.
\end{equation*}
\end{lemm}

\begin{proof}
We take $p\geq 2$ and use from~\eqref{yazahraadrecni} and integration by parts to obtain:
\begin{align}\label{313***}
&\frac{\dd}{\dt}\int_{\Omega} u^{p} \,\dx= p\int_{\Omega} u^{p-1} \Big[\nabla\cdot(\gamma(v) \nabla u-u\, \xi(v)\nabla v)+ \mu u (1- u) \Big]\, \dx \nonumber\\
&\quad =-p(p-1)\int_{\Omega}\gamma(v)\, u^{p-2} | \nabla u|^{2} \, \dx + p(p-1) \int_{\Omega} u^{p-1} \xi(v)\, \nabla u\cdot \nabla v \, \dx  + \mu p \int_{\Omega} u^{p} (1- u) \, \dx.
\end{align}
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}:=c_{1},\\
\xi(v)&=k(1-\alpha)(1+v)^{-k-1}\leq k(1-\alpha):=c_{2}.
\end{align*}
Making use of these, \eqref{313*****} and Young's inequality, we can write~\eqref{313***} as follows:
\begin{align}\label{ya mahdi}
\frac{\dd}{\dt}\int_{\Omega} u^{p} \,\dx &\leq- c_{1}\,p\,(p-1)\int_{\Omega} u^{p-2} | \nabla u|^{2} \, \dx + C \,c_{2} \,p\,(p-1)\int_{\Omega} u^{p-1} |\nabla u| \,\dx +\mu p\int_{\Omega} u^{p}\, \dx\nonumber\\
&\quad = -\frac{4 \,c_{1}(p-1)}{p}\int_{\Omega}\big|\nabla u^{\frac{p}{2}}\big|^{2} \, \dx +2\, C \,c_{2} \,(p-1)\int_{\Omega} u^{\frac{p}{2}}\cdot \big|\nabla u^{\frac{p}{2}}\big| \, \dx +\mu p\int_{\Omega} u^{p}\, \dx\nonumber\\
&
\leq-\frac{2\,c_{1}\,(p-1)}{p}\int_{\Omega}\big|\nabla u^{\frac{p}{2}}\big|^{2} \dx+p\,\Biggl(\frac{(p-1)\,C^{2}c_{2} ^{2}}{2\,c_{1}}+\mu \Biggr) \int_{\Omega} u^{p}\dx.
\end{align}
We now add $p \int_{\Omega} u^{p}\, \dx$ on both sides of~\eqref{ya mahdi} to have:
\begin{equation}\label{main essential}
\frac{\dd}{\dt}\int_{\Omega} u^{p} \dx + p \int_{\Omega} u^{p}\, \dx \leq -\frac{2\,c_{1}\,(p-1)}{p}\int_{\Omega}|\nabla u^{\frac{p}{2}}|^{2} \dx + c_{3}\int_{\Omega} u^{p}\, \dx
\end{equation}
with
\begin{equation*}
c_{3}=p\,\Biggl(\frac{(p-1)\,C^{2}c_{2} ^{2}}{2\,c_{1}}+\mu +1\Biggr).
\end{equation*}
To estimate the last term on the right hand side of~\eqref{main essential}, we use the following known Gagliardo--Nirenberg inequality (see~\cite{hw, WZ}, for instance):
\begin{align*}
\big\|\psi \big\|_{L^{q}(\Omega)} &\leq C_{GN}\Big(\big\|\nabla \psi\big\|_{L^{2}(\Omega)}^{\vartheta}\big\| \psi\big\|_{L^{r}(\Omega)} ^{1-\vartheta}+\big\| \psi \big\|_{L^{r}(\Omega)}\Big),
\end{align*}
where
\begin{align*}
q(n-2)<2n, \quad r\in (0,q) \quad \text{and}\quad
\vartheta=\frac{\frac{n}{r}-\frac{n}{q}}{1-\frac{n}{2}+\frac{n}{r}}\in (0,1),
\end{align*}
and $C_{GN}$ is the constant in the Gagliardo--Nirenberg inequality. Now, we apply the Gagliardo--Nirenberg inequality with $\psi=u^{\frac{p}{2}}, q=2, r=1$ and $\vartheta=\frac{n}{n+2},$ and then use the Young inequality with exponents $r=\frac{n+2}{n}$ and $s=\frac{n+2}{2}$ to obtain:
\begin{align*}
c_{3}\int_{\Omega} u^{p}\, \dx =c_{3} \, \big\|u^{\frac{p}{2}}\big\|_{L^{2}(\Omega)}^{2} &\leq c_{3} \,(C_{GN})^{2}\Big(\big\|\nabla u^{\frac{p}{2}}\big\|_{L^{2}(\Omega)}^{\frac{n}{n+2}}\big\| u^{\frac{p}{2}}\big\|_{L^{1}(\Omega)}^{\frac{2}{n+2}}+\big\| u^{\frac{p}{2}}\big\|_{L^{1}(\Omega)}\Big)^{2}\\
&\leq 2\, c_{3} \,(C_{GN})^{2}\Big(\big\|\nabla u^{\frac{p}{2}}\big\|_{L^{2}(\Omega)}^{\frac{2n}{n+2}}\big\| u^{\frac{p}{2}}\big\|_{L^{1}(\Omega)}^{\frac{4}{n+2}}+\big\| u^{\frac{p}{2}}\big\|_{L^{1}(\Omega)}^{2}\Big)\\
&\leq
\frac{2\,c_{1}\,(p-1)}{p}\big\|\nabla u^{\frac{p}{2}}\big\|_{L^{2}(\Omega)}^{2}+\big(c_{4}+2\, c_{3}\, (C_{GN})^{2}\big)\big\| u^{\frac{p}{2}}\big\|_{L^{1}(\Omega)}^{2}\\
& =\frac{2\,c_{1}(p-1)}{p}\int_{\Omega}|\nabla u^{\frac{p}{2}}|^{2}\, \dx +c_{5}\Big(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Big)^{2}
\end{align*}
with
\begin{align*}
c_{4}=\frac{1}{s}\Biggl(\frac{2\,c_{1}\, r\, (p-1)}{p}\Biggr)^{-\frac{s}{r}}\big(2\, c_{3} \,(C_{GN})^{2}\big)^{s}
\quad \text{and}\quad c_{5}=c_{4}+2\, c_{3} (C_{GN})^{2}.
\end{align*}
Combining the last inequality with~\eqref{main essential} yields:
\begin{equation*}
\frac{\dd}{\dt}\int_{\Omega} u^{p} \, \dx +p \int_{\Omega} u^{p}\, \dx \leq c_{5} \Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{2}.
\end{equation*}
For $0 \leq t \leq T_{\max}, $ we can write:
\begin{equation*}
\frac{\dd}{\dt}\Biggl(\eit^{p\,t}\int_{\Omega} u^{p} \, \dx \Biggr) \leq c_{5}\, \eit^{p\,t}\,
\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{2}.
\end{equation*}
Now, we integrate and use $\eit^{-p\,t}\leq 1$ to get:
\begin{align*}
\int_{\Omega} u^{p}\, \dx &
\leq \int_{\Omega} u^{p}_{0}\, \dx+
\frac{c_{5}}{p} \sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{2}\nonumber\\
&
\leq |\Omega|\, \| u_{0}\|_{L^{\infty}(\Omega)}^{p}+
\frac{c_{5}}{p} \sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{2}.
\end{align*}
Thus,
\begin{align}\label{main essential11}
\Biggl(\int_{\Omega} u^{p} \,\dx\Biggr)^{\frac{1}{p}} &
\leq \Biggl[\lvert\Omega|\, \| u_{0}\|_{L^{\infty}(\Omega)}^{p}+
\frac{c_{5}}{p} \sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{2}\Biggr]^{\frac{1}{p}}\nonumber\\
&
\leq |\Omega|^{\frac{1}{p}}\, \| u_{0}\|_{L^{\infty}(\Omega)}+
\big(\frac{c_{5}}{p}\big)^{\frac{1}{p}} \sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{\frac{2}{p}}.
\end{align}
We note that
\begin{align*}
c_{5}&=c_{4}+2\, c_{3} (C_{GN})^{2} \\
&=\frac{1}{s}\Biggl(\frac{2\,c_{1}\, r\, (p-1)}{p}\Biggr)^{-\frac{s}{r}}\big(2\, c_{3} \,(C_{GN})^{2}\big)^{s}+2\, c_{3} (C_{GN})^{2}\\
&=\frac{1}{s}\big(2\,c_{1}\, r\big)^{-\frac{s}{r}}\big(2
\,(C_{GN})^{2}\big)^{s}
\Biggl(\frac{p}{p-1}\Biggr)^{\frac{n}{2}}(c_{3})^{s}+2\, c_{3}\, (C_{G N})^{2}\\
& \leq m \Biggl[\Biggl(\frac{p}{p-1}\Biggr)^{\frac{n}{2}}(c_{3})^{s}+c_{3} \Biggr]\\
& \leq m \Biggl[\Biggl(\frac{p}{p-1}\Biggr)^{\frac{n}{2}}+1 \Biggr](c_{3})^{s}
\end{align*}
with
\begin{equation*}
m=\max\Biggl\{\frac{1}{s}\,\big(2\,c_{1}\, r\big)^{-\frac{s}{r}}\big(2
\,(C_{GN})^{2}\big)^{s},2\, (C_{GN})^{2}\Biggr\}.
\end{equation*}
Here, we have used from $c_{3}>1$ and $s>1.$ By inserting $c_{3}$ and using $p \geq 2,$ we obtain:
\begin{align}\label{salam}
\frac{c_{5}}{p}&\leq m \Biggl[\Biggl(\frac{p}{p-1}\Biggr)^{\frac{n}{2}}+1 \Biggr]\Biggl(\frac{(p-1)\,C^{2}c_{2} ^{2}}{2\,c_{1}}+\mu +1\Biggr)^{\frac{n}{2}+1} p^{\frac{n}{2}}\nonumber\\
& \leq 2\, m \,\Biggl(\frac{ C^{2}c_{2} ^{2}}{2\,c_{1}}+\mu +1\Biggr)^{\frac{n}{2}+1}\, \Biggl(\frac{p}{p-1}\Biggr)^{\frac{n}{2}} (p-1)^{\frac{n}{2}+1}\, p^{\frac{n}{2}}\nonumber\\
&= c_{6} \, (p-1)\, p^{n}\nonumber\\
& \leq c_{6} \, p^{n+1}
\end{align}
with
\begin{align*}
c_{6}= 2\, m \,\Biggl(\frac{ C^{2}c_{2} ^{2}}{2\,c_{1}}+\mu +1\Biggr)^{\frac{n}{2}+1}.
\end{align*}
Making use of~\eqref{salam} and $p^{\frac{n+1}{p}}>1,$ we can write~\eqref{main essential11} as follows:
\begin{align}\label{yalatif}
\Biggl(\int_{\Omega} u^{p} \,\dx\Biggr)^{\frac{1}{p}} &
\leq |\Omega|^{\frac{1}{p}}\, \| u_{0}\|_{L^{\infty}(\Omega)}+
\big(c_{6} \, p^{n+1}\big)^{\frac{1}{p}} \sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{\frac{2}{p}}\nonumber\\
&
\leq c_{7}^{\frac{1}{p}} \, p^{\frac{n+1}{p}} \Biggl(\| u_{0}\|_{L^{\infty}(\Omega)}+
\sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{\frac{p}{2}}\, \dx\Biggr)^{\frac{2}{p}}\Biggr)
\end{align}
with $c_{7}=|\Omega|+ c_{6}.$ We now define:
\begin{equation*}
M(p)=\max\Biggl\lbrace \| u_{0}\|_{L^{\infty}(\Omega)}, \sup_{0\leq t\leq T_{\max}}\Biggl(\int_{\Omega} u^{p}\dx\Biggr)^{\frac{1}{p}}\Biggr\rbrace.
\end{equation*}
This allows us to write~\eqref{yalatif} as:
\begin{equation*}
M(p)\leq 2\, c_{7}^{\frac{1}{p}} \, p^{\frac{n+1}{p}} M\biggl(\frac{p}{2}\biggr).
\end{equation*}
We now take $p= 2^{i}$ $(i\in\mathbb{N})$ to obtain:
\begin{align}\label{yakarim}
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 \vphantom{2^{\big(\frac{i}{2^{i}}\big)}}\\
& \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{align}
We now compute the following elementary series
\begin{align*}
S:= \sum_{i=1}^{\infty}\frac{i}{2^{i}}&= \sum_{i=0}^{\infty}\frac{i+1}{2^{i+1}}=\sum_{i=0}^{\infty}\Biggl(\frac{i}{2^{i+1}}+\frac{1}{2^{i+1}}\Biggr)=\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(\,\cdot\,,t)\|_{L^{2^{i}}(\Omega)}=\|u(\,\cdot\,,t)\|_{L^{\infty}(\Omega)} $ and~\eqref{mamnonam}, by letting $i \rightarrow \infty$ in~\eqref{yakarim}, we obtain the desired result.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm}]
By considering the extensibility criterion provided by Lemma~\ref{lem3}, the proof is a consequence of~\eqref{yaallah} and Lemma~\ref{lem4}.
\end{proof}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referees for their careful reading and valuable suggestions on this article.

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