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\title[Blow--up for system of $\kappa$th order wave equations]{Finite time blow--up  of solutions for system of $\kappa$th order wave equations with nonlinear averaged damping and sources}


\author{K\lowercase{h}. Zennir, K. Bouhali, H. Zhang and D. Guidad}

\address{Khaled Zennir,
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} College of Science,
\newline\hphantom{iii} Qassim University,
\newline\hphantom{iii} Buraidah, Saudi Arabia.
}
\email{k.zennir@qu.edu.sa}

\address{Keltoum Bouhali,
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} College of Science,
\newline\hphantom{iii} Qassim University,
\newline\hphantom{iii} Buraidah, Saudi Arabia.
}
\email{k.bouhali@qu.edu.sa}


\address{Hongwei Zhang,
\newline\hphantom{iii} School of Mathematics and Statistics,
\newline\hphantom{iii} Henan
University of  Technology,
\newline\hphantom{iii} Zhengzhou 450001, China.
}
\email{whz661@163.com}

\address{Derradji Guidad,
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} College of Sciences,
\newline\hphantom{iii} University of Biskra,
\newline\hphantom{iii} BP 145 RP, 07000 Biskra,  Algeria.}
\email{guidadd07@gmail.com}


\thanks{\textsc{Kh. Zennir, K. Bouhali, H. Zhang and D. Guidad, Finite time blow--up  of solutions for   %nonlinear coupled
system of $\kappa$th order wave equations with nonlinear averaged damping and sources}}
\thanks{\textcopyright\ Zennir Kh., Bouhali K., Zhang H. and Guidad D.  2026}
\thanks{\it Submitted February 27, 2025.}

\maketitle {\small
\begin{quote}
\noindent\noindent{\bf Abstract.}   We consider basic relationships between the initial energy and the exponent of nonlinear sources in a nonlinear coupled system of $\kappa$th order wave equations with nonlinear averaged damping, and we demonstrate the global nonexistence of solutions. This approach is a variant of the method of nonlinear functional analysis  with a contradiction argument, which is one of the tools for proving the blow--up of solutions for nonlinear partial differential equations.    A new class of nonlinear coupled wave equations with nonlinear sources is given in high-order functional spaces.   %The developed approach allows for blow--up results.




\medskip



\noindent{\bf Keywords:} $\kappa$th order wave equations, nonlinear equations, coupled system, nonlinear averaged damping, local existence,  blow--up.

\medskip
\noindent{\bf Mathematics Subject Classification: }{35A01; 35L20; 35B44}

\end{quote}
}


\vskip30pt

\section{Introduction, relevance and objectives }
Let $\Omega \subset \mathbb{R}^n$ $(n\ge 1)$ be a bounded domain with a smooth boundary $\Gamma$.
%Let $u=u(x, t), v=v(x, t)$, for $x\in \Omega, t \in [0, +\infty)$,
We consider the initial boundary value problem for nonlinear $\kappa$th order wave equations
with a\ nonlocal damping
\begin{equation}\label{1.1}
\left\{
\begin{aligned}
&u_{tt}+ (-\Delta)^\kappa u
+\|u_t\|^{l}u_t=|u|^{p-2}u\vert v\vert ^p,
\\
&v_{tt}+ (-\Delta)^\kappa v
+\|v_t\|^{l}v_t=|v|^{p-2}v\vert u\vert ^p,
\end{aligned}
\right. \qquad x\in\Om,\quad t>0,
\end{equation}
with boundary conditions
\begin{equation}  \label{1.2}
\left\{
\begin{aligned}
u=\frac{\partial u}{\partial \nu}=\ldots=\frac{\partial^{\kappa -1} u}{\partial \nu^{\kappa -1}}=0,
 \\
v=\frac{\partial v}{\partial \nu}=\ldots=\frac{\partial^{\kappa -1} v}{\partial \nu^{\kappa -1}}=0,
\end{aligned}
\right.\qquad x\in  \Gamma,
\end{equation}
and initial conditions
\begin{equation} \label{1.3}
\left\{
\begin{aligned}
&u(x,0) = u_0(x),\qquad u_t(x,0) = u_1(x),
\\
&v(x,0) = v_0(x),\hspace{3 true pt}\qquad v_t(x,0) = v_1(x),
\end{aligned}
\right.\qquad x\in\Om.
\end{equation}
Here $\nu $ is the outward normal to the boundary $\G,$ while  $l>0,$ $\kappa\geq 1$ are integers and $p>2$, and  for $u=u(x, t)$
\begin{align*}
&\vert\nabla^{\kappa}u\vert^{2}=(\Delta^{\frac{\kappa}{2}}u)^{2}\hspace{26 true pt}\qquad \text{for even}\quad  \kappa,
\\
&\vert\nabla^{\kappa}u\vert^{2}=\vert\nabla(\Delta^{\frac{\kappa-1}{2}}u ) \vert^{2}
\qquad \text{for even}\quad   \kappa,
\end{align*}
and
\begin{equation*}
\vert\nabla u\vert^{2}= \sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_{i}}\right)^{2},
\qquad \Delta u = \sum_{i=1}^{n}\frac{\partial^{2} u}{\partial x_{i}^{2}}.
\end{equation*}
Recently, an initial boundary value problem for following wave equation with nonlinear averaged damping
\begin{align}
&u_{tt}- \Delta  u+g\Big(\int\limits_{\Omega}|u_t|^2\,dx\Big)u_t=f(u) \label{1.4}
\end{align}
attracted a lot of attention. Equation \eqref{1.4} with the source $f(w)=w\int\limits_{\Omega}|w|^2\,dx$ arises from a nonlinear theory of meson fields \cite{Jorgens1961}, \cite{Louredo2014}, \cite{Schiff1951}.  It can also be intrinsically connected with the
study of damping phenomena in flight structures, where the   $\|u_t \|^m$
reflects the effect of kinetic energy; see   \cite{Aljurbua2025}, \cite{Silva2019DCDS}, \cite{Zennir2024}.
For instance, in  \cite{Balakrishnan1989}, the   one dimensional beam equation
\begin{align}
 u_{tt}-2\varsigma\sqrt{\lambda}u_{xx}+\lambda u_{xxxx}
-\gamma\Bigg(\int\limits_{L}^{-L}(\lambda| u_{xx}|^2+| u_{t}|^2\,dx\Bigg)^q u_{xxt}=0 \label{1.5}
\end{align}
was proposed.
As the first stage of the corresponding equation
$$u_{tt}+\Big(\int\limits_\Omega u_t^2\,dx\Big)^\frac{\alpha}{2}u_t-\Delta u=h(t,x),$$
the pioneering works \cite{Aloui2013JMPA} provided the above equation in a bounded domain $\Omega$
with homogeneous boundary conditions; see  \cite{Aloui2013JMPA}, \cite{Ghisi2019ANA}, \cite{zennir2018}. Later, the authors in \cite{Louredo2014} studied the system
\begin{equation}
\begin{aligned}
&u_{tt}- \mu(t)\Delta u+\alpha f\bigg(\int\limits_\Omega |u|^{2}\,dx\bigg)u+\beta g\bigg(\int\limits_\Omega |u_t|^{2}\,dx\bigg)u_t
=0,  \\
&u=0\quad\text{on}\quad \Gamma_0,
\qquad \frac{\partial u}{\partial \nu} +h(\,\cdot\,,u_t)=0\quad\text{on}\quad \Gamma_1,  \\
&u(x, 0) = u_0(x),\qquad u_t(x,0) = u_1(x),
\end{aligned}
\end{equation}
under certain conditions on $\mu(t),$ $ f(s), $ $ g(s), $ $ h(s)$. Global
existence in time was shown by using the well--known Galerkin method. In addition,  a decay rate of the solution was obtained by two different methods, both Lyapunov functional and Nakao's lemma.
Zhang \cite{Zhang2019JPDE} extended and improved
the result in \cite{Louredo2014}. In \cite{Zhao2020DCDS}, \cite{Zhao2023ActaMS}, the long time behavior of the  equations
\begin{align}
&u_{tt}+\Delta ^2u-M(\|\nabla u\|^{2}) \Delta u+k(\|u_t\|)u_t+f(u)=h(x) \label{1.10}
\\
&u_{tt}-\Delta u+k(\|u_t\|)u_t+f(u)=\int\limits_{\Omega}K(x,y)u_t(y)dy+h(x) \label{1.11}
\end{align}
was considered.
The global well--posedness of \eqref{1.10} \eqref{1.11} was established  with the  damping $\|u_t\|^m u_t,$ $m\ge 0,$  see
\cite{Li2022EECT}, \cite{Peng2023AMO}, \cite{Zhang2022AA} and references therein. In \cite{Silva2019DCDS}, the authors investigated the nonhomogeneous $n$--dimensional version of \eqref{1.5}
\begin{equation}
{u_{tt}} - \kappa{\Delta }u +{\Delta^2 }u-  \gamma \bigg(\int\limits_\Omega  {({{\left| {\Delta u} \right|}^2} + {{\left| {{u_t}} \right|}^2})\,dx} \bigg)^q\Delta {u_t}+f(u) =0  \quad\text{in}\quad \Omega\times {R}^+ \label{1.12}
\end{equation}
 subject to a boundary condition and with $q\geq1$. The authors proved the existence of  a unique global solution and obtained the polynomial stability and a non-exponential decay prospect of the
solutions provided that $q\geq1,$ see  \cite{Chueshov2008AMS}, \cite{Yang2022JMAA}, \cite{Zennir2015}.


 For the equation
\begin{align}
&u_{tt}- \Delta u+ \bigg(\int\limits_\Omega |u_t|^{2}\,dx\bigg)^\frac{m}{2}u_t= \bigg(\int\limits_\Omega |u|^{p}\,dx\bigg)^r|u|^{p-2}u \label{1.13}
\end{align}
with homogeneous boundary conditions, Hu, Li, Liu and Zhang  \cite{Zhang2023QM}  provided
sufficient conditions for finite time blow--up of weak solutions, with suitable conditions
on initial data, by an ordinary differential inequality for an appropriately chosen functional.
When the terms $\big(\int\limits_\Omega |u|^{p}\,dx\big)^r|u|^{p-2}u$ and $-\Delta u$ in equation \eqref{1.13}
are replaced by $|u|^{p-2}u $ and $ \Delta^2 u$, respectively,
 Zennir and Miyasita \cite{Miyasita2020} proved the global non-existence in time for a large class of pseudo-parabolic equations with weak viscoelasticity under suitable conditions on the variable exponents with negative initial energy.


To our knowledge, there are few results on the non-linear coupled system of $\kappa$th order wave equations with nonlinear averaged damping and sources. The main goal of this paper is to study the blow--up conditions of a solution to problem \eqref{1.1}--\eqref{1.3} with the averaged damping term and the source term, which differ from the existing literature mentioned above.  To the best of our knowledge, this is the first result on the nonlinear coupled system of $\kappa$th order
wave equations with nonlinear averaged damping and sources.


The paper is organized as follows. Section 2 is concerned with some notation and necessary materials.  In Section 3, we demonstrate the global nonexistence of solutions with positive or negative initial energy.

\vskip20pt

\section{Preliminaries}

Let  $L^2(\Omega),H^p(\Omega)$ and $H^2_0(\Omega)$ be the usual Sobolev spaces, where
\begin{equation*}
(u,v)=\int\limits_{\Omega}uv\,dx, \qquad \|u\|=\bigg(\int\limits_{\Omega}|u|^2\,dx\bigg)^\frac{1}{2},\qquad
\|u\|_p
=\bigg(\int\limits_{\Omega}|u|^p\,dx\bigg)^\frac{1}{p}.
\end{equation*}

\begin{definition}
Functions $$(u, v)\in  C\Big([0, T]; H ^\kappa(\Omega)\Big) \cap C^1 \Big([0, T]; L^2 (\Omega)\Big)\times  C\Big([0, T]; H ^\kappa(\Omega)\Big) \cap C^1 \Big([0, T]; L^2 (\Omega)\Big),$$
with the initial data given in \eqref{1.3} are

\vskip5pt

\begin{enumerate}
    \item  a strong solution to problem \eqref{1.1}--\eqref{1.3} on the interval $[0, T]$
 if
 \begin{align*}
 &(u,v) \in W^{1,1}(\alpha, \beta; H^\kappa_0(\Omega)) \times W^{1,1}(\alpha, \beta; H^\kappa_0(\Omega)),
 \\
 &(u_t, v_t)\in W^{1,1}(\alpha, \beta; L^2(\Omega)) \times W^{1,1}(\alpha, \beta; L^2(\Omega)),
 \end{align*}
 for all $0 < \alpha < \beta < T$ and $$\Delta^\kappa u + \Delta^\kappa v + \|u_t(t)\|^l u_t + \|v_t(t)\|^l v_t \in L^2(\Omega)\quad \text{for a.e.}\quad t\in [0,T],$$
  and \eqref{1.1} is satisfied in $L^2 (\Omega)$ for all $t\in [0, T];$

\vskip5pt

\item  a generalized solution to \eqref{1.1}-\eqref{1.3} on the interval $[0, T]$
  if there is a sequence of strong solutions $(\{u_j (t)\}, \{v_j (t)\})$ to \eqref{1.1}-\eqref{1.3} with initial data $(u^j_0, v^j_0), (u^j_1, v^j_1)$ instead of $(u_0, v_0 ), (u_1, v_1 )$
such that
\begin{align*}
&(u^j, v^j )\rightarrow  (u, v)\quad \hspace{6.5 true pt}\text{in}\quad C([0, T]; H^\kappa_0(\Omega))\times C([0, T]; H^\kappa_0(\Omega)) \quad \text{as} \quad j \rightarrow +\infty,
\\
&(u^j_t, v^j_t)\rightarrow  (u_t, v_t ) \quad \text{in}\quad C([0, T]; L^2(\Omega))\times C([0, T]; L^2(\Omega))\hspace{6.5 true pt}\quad \text{as} \quad j \rightarrow +\infty.
\end{align*}

\vskip5pt

\item  a distributed solution to \eqref{1.1}--\eqref{1.3} on $[0, T]$  if
\begin{equation}
\begin{aligned}
\int\limits_{\Omega}&\big(u_{t}(x,t)\phi (x)+v_{t}(x,t)\psi(x)\big)\,dx
\\
&+\int\limits_0^t\Bigg(\int\limits_{\Omega} \nabla^\kappa u(x,s)\nabla^\kappa \phi(x)\,dx
+\|u_t(s)\|^l\int\limits_{\Omega} u_t(x,s)\phi(x)\,dx\Bigg)\,ds
\\
&+\int\limits_0^t\Bigg(\int\limits_{\Omega} \nabla^\kappa v(x,s)\nabla^\kappa \psi(x)\,dx
+\|v_t(s)\|^l\int\limits_{\Omega} v_t(x,s)\psi(x)\,dx\Bigg)\,ds
\\
=&\int\limits_{\Omega}u_1\phi (x)\,dx+\int\limits_0^t\int\limits_{\Omega}|u|^{p-2}u(x,s)|v(x, s)|^p\phi(x)\,dxds
\\
&+\int\limits_{\Omega}v_1\psi (x)\,dx+\int\limits_0^t\int\limits_{\Omega}|v|^{p-2}v(x,s)|u(x, s)|^p\psi(x)\,dxds
\end{aligned}\label{2.1}
\end{equation}
holds for every test function $\phi, \psi \in H^\kappa_0(\Omega)\cap L^2(\Omega)$ and for all $t\in [0, T].$
\end{enumerate}
\end{definition}


\begin{lemma}\label{lemma2.2} \cite{Zhao2020DCDS} For $u_t, w_t\in L^2(\Omega)$, we have
\begin{align}
& \|u_t-w_t\|^{l+2} \leq \left(\|u_t\|^lu_t-\|w_t\|^lw_t,u_t-w_t\right).\label{2.2}
\end{align}
\end{lemma}
The associated energy of  \eqref{1.1}-\eqref{1.3} is defined  by
\begin{align}
&E(t)=\frac{1}{2}\|u_t\|^2+\frac{1}{2}\|v_t\|^2+J(u, v),\quad 0\le t,\label{2.3}\\
&J(u, v)=\frac{1}{2}\|\nabla^\kappa u\|^2+\frac{1}{2}\|\nabla^\kappa v\|^2-\frac{1}{p}\|uv\|_p^p,\label{2.4}
\end{align}
and
\begin{align}
&E|_{t=0}=\frac{1}{2}\|u_1\|^{2}+\frac{1}{2}\|v_1\|^{2}+J(u_0, v_0),\label{2.5}
\\
&\|u\|_p\le C_0\|\nabla^\kappa u\|, \ C_0>0, \ \forall u\in H_0^\kappa(\Omega),\nonumber
\end{align}
 where
\begin{equation}
\left\{
\begin{aligned}
&2 \leq p <+\infty && \text{if}\quad  n=\kappa,2\kappa
 \\
&2 \leq p \leq \frac{2n}{n-2\kappa} && \text{if}\quad n\geq 3\kappa.
\end{aligned}
\right.  \label{pcondition}
\end{equation}

\vskip20pt

\section{Finite time blow up of solutions}

We state the local existence in time of \eqref{1.1}--\eqref{1.3} by following the lines of
 \cite{Chueshov2008AMS}.

\begin{theorem} \label{Theoremlocal} If $(u_0, v_0) \in (H^{2\kappa}(\Omega) \cap H^\kappa_0(\Omega))\times (H^{2\kappa}(\Omega) \cap H^\kappa_0(\Omega))$
and $(u_1, v_1) \in H^\kappa_0(\Omega) \times H^\kappa_0(\Omega)$ are such
that $$\nabla^\kappa u_0 +\nabla^\kappa v_0 +k\|u_1 \|^l u_1+k\|v_1 \|^l v_1 \in L^2 (\Omega).$$
Suppose that $l> 0$ and $p$ satisfies
\begin{equation} \label{c3.1}
\left\{
\begin{aligned}
&1 \leq p <+\infty &&  \text{if}\quad n=\kappa,2\kappa  \\
&1 \leq p \leq \frac{4\kappa-n}{n-2\kappa}\quad &&  \text{if}\quad  n\geq 3\kappa.
\end{aligned}
\right.
\end{equation}
Then  there exists $t_{\mathrm{max}}\le +\infty$
such that  the problem  \eqref{1.1}--\eqref{1.3}
has a unique strong solution $(u, v)$
on $[0, t_{\mathrm{max}})$.


If $(u_0, v_0) \in H^\kappa_0(\Omega)\times H^\kappa_0(\Omega)$ and $(u_1, v_1) \in L^2(\Omega)\times L^2(\Omega)$ ,
then there exists $t_{\mathrm{max}}\le +\infty$
such that \eqref{1.1}--\eqref{1.3} admits a
unique generalized solution,
which is also the distributional solution.


With positive and negative initial data, both the strong solution and the weak solution $(u, v)$
 cannot be global solutions in $[0, t_{max} ), \ t_{\mathrm{max}} = +\infty$,   so for
$t_{\mathrm{max}} < \infty$ we have
\begin{align}
&\lim_{t\rightarrow t_{\mathrm{max}}}\left(\|\nabla^\kappa u\|+\|\nabla^\kappa v\|+\|u_t\|+\|v_t\|\right)=+\infty,\label{3.1}
\end{align}
 The energy relation is satisfied
\begin{align}
&E(t)+\int\limits_0^t(\|u_t(s)\|^{l+2}+\|v_t(s)\|^{l+2})ds=E(0),\quad 0\le t<t_{\mathrm{max}}.\label{3.2}
\end{align}
\end{theorem}

\vskip10pt

\subsection{Negative initial energy} Our first main result is as follows.

\begin{theorem} \label{theoremblowup}
Let $E(0) < 0$. Under the assumptions of Theorem~\ref{Theoremlocal} there exists a finite time $t_{\mathrm{max}}<+\infty,$ at which the local solution to \eqref{1.1}--\eqref{1.3} blows up.
\end{theorem}

\begin{proof}
Let $$\mathcal{K}(t)=-E(t).$$ The contradiction argument will be used. By \eqref{2.4} and \eqref{3.2}  we obtain
\begin{align}
& \mathcal{K}'(t)=-E'(t)\ge 0, \nonumber
\\
&0<- E(0)= \mathcal{K}(t=0)\le \mathcal{K}(t)\le \frac{1}{p}\|uv\|^p_p.\label{5.3}
\end{align}
We denote
$$G(t)=\int\limits_{\Omega}(\|u_t\|^lu_tu+\|v_t\|^lv_tv)\,dx,$$
then, by H\"older and $\varpi$-Young's inequalities for each $\varpi>0$ we obtain
\begin{align*}
&
\|u_t\|^2\,\|u\|^2
\le \varepsilon \|u_t\|^{2} + \frac{1}{4\varepsilon}\|u\|^{2},
\\
&|G(t)|\le \|u_t\|^l\|u_t\|\|u\|+\|v_t\|^l\|v_t\|\|v\|
\\
&\hphantom{|G(t)|}\leq \|u_t\|^l\left(  \varpi \|u_t\|^2+\frac{1}{4\varpi} \|u\|^2\right) +  \|v_t\|^l\left( \varpi  \|v_t\|^2+\frac{1}{4\varpi}  \|v\|^2\right)
\\
&\hphantom{|G(t)|}\leq \varpi[\|u_t\|^{l+2}+\|v_t\|^{l+2}]+ \frac{1}{4\varpi}  [\|u_t\|^l\|u\|^2 +\|v_t\|^l\|v\|^2]
\end{align*}
Again, using Young's inequality with
$$
p=\frac{l+2}{l}, \qquad q=\frac{l+2}{2}, \qquad \frac{1}{p}+\frac{1}{q}=1,
$$
for each  $\varpi>0$ we obtain,
\begin{equation*}
\|u_t\|^{l}\|u\|^{2}\leq  \varpi\|u_t\|^{lp}
+ \varpi^{-\frac{2}{l}}\|u\|^{2q} =  \varpi \|u_t\|^{l+2}
+ \varpi^{-\frac{2}{l}}\|u\|^{l+2},
\end{equation*}
and similarly,
$$
\|v_t\|^{l}\|v\|^{2}
\leq \varpi \|v_t\|^{l+2}
+ \varpi^{-\frac{2}{l}}\|v\|^{l+2}.
$$
Hence,
 $$
 \Vert u_t\Vert ^l \Vert u\Vert ^2+ \Vert v_t\Vert ^l \Vert v\Vert ^2 \leq \varpi (\Vert u_t\Vert ^{l+2} +\Vert v_t\Vert ^{l+2} )+\varpi^{-\frac{2}{l}}(\Vert u\Vert ^{l+2} +\Vert v\Vert ^{l+2} ),
 $$
 which leads by \eqref{3.2} to
\begin{equation} \label{5.5}
\begin{aligned}
 |G(t)| \leq&  (\varpi+\frac{1}{4})[\|u_t\|^{l+2}+\|v_t\|^{l+2}]+\frac{ \varpi^{-\frac{2+l}{l}}}{4 }(\Vert u\Vert ^{l+2} +\Vert v\Vert ^{l+2} )  \\
 \leq& (\varpi+\frac{1}{4})\mathcal{K}'(t) +\frac{\varpi^{-\frac{2+l}{l}}}{4} (\Vert u\Vert ^{l+2} +\Vert v\Vert ^{l+2} ).
 \end{aligned}
\end{equation}
We define the functional
\begin{align}
&z(t)=\mathcal{K}^{1-\nu}(t)+\epsilon \int\limits_{\Omega} u_{t}u\,dx+v_{t}v\,dx, \qquad \epsilon>0,\qquad 0<\nu<1. \label{5.6}
\end{align}
Taking the derivative of \eqref{5.6} in $t$,
by \eqref{1.1}, and \eqref{5.5}, we have
\begin{equation}
\begin{aligned}
 z'(t) =& (1-\nu)\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)
+\epsilon \int\limits_{\Omega} (u_{tt}u+v_{tt}v)\,dx+\epsilon (\|u_t\|^2+ \|v_t\|^2)
\\
 =& (1-\nu)\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)
+\epsilon \Big(\|u_t\|^2+\|v_t\|^2- \|\nabla^\kappa u\|^2- \|\nabla^\kappa v\|^2+\|uv\|_p^p-G(t)\Big)
 \\
 \geq&   \bigg((1-\nu)\mathcal{K}^{-\nu}(t)-\frac{1}{4}\epsilon \varpi \bigg)\mathcal{K}'(t)
+\epsilon \big(\|u_t\|^2+\|v_t\|^2- \|\nabla^\kappa u\|^2- \|\nabla^\kappa v\|^2+\|uv\|_p^p \big)
 \\
 -&  \frac{\varpi^{-\frac{2+l}{l}}  }{4} \big(\|u\|^{l+2}+\|v\|^{l+2}\big).\label{5.7}
\end{aligned}
\end{equation}
We can choose the Young parameter $\varpi$ as a time–dependent weight of $\mathcal K(t)$, namely
$$
\varpi = k\,\mathcal K^{-\nu}(t), \qquad k>0 .
$$
Moreover,
\begin{equation*}
\varpi^{-\frac{l+2}{l}}
 = k^{-\frac{l+2}{l}}\,
\mathcal K^{\frac{\nu(l+2)}{l}}(t),
\end{equation*}
where the constant $k$ is redefined if necessary.


Substituting these estimates into \eqref{5.7}, we obtain
\begin{align*}
z'(t)\ge &
\Big(1-\nu-\epsilon\frac{k}{4}\Big)
\mathcal K^{-\nu}(t)\mathcal K'(t)
\\
&
+\epsilon\bigg(
\|u_t\|^2+\|v_t\|^2
-\|\nabla^\kappa u\|^2-\|\nabla^\kappa v\|^2
+\|uv\|_p^p
-\frac{1}{4k^{\frac{l+2}{l}}}\mathcal K^{ \frac{\nu(l+2)}{l}}(t)
\big(\|u\|^{l+2}+\|v\|^{l+2}\big)
\bigg),
\end{align*}
We take
\begin{equation} \label{alpha}
0 \le \alpha < \frac{p-2}{p}.
\end{equation}
Then,
\begin{equation}
\begin{aligned}
 z'(t)
\geq& \bigg(1-\nu-\epsilon  \frac{k}{4}\bigg)
\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)+\epsilon p(1-\alpha)\mathcal{K}(t)+\epsilon \bigg(\frac{ p(1-\alpha)}{2}+1\bigg) (\|u_t\|^2+\|v_t\|^2)
 \\
+&\epsilon \left(\frac{ p(1-\alpha)}{2}-1\right) \big(\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2\big)
+\epsilon p\|uv\|_p^p
\\
&- \epsilon \frac{1}{4k^{\frac{l+2}{l}}}\mathcal K^{ \frac{\nu(l+2)}{l}}(t) \big(\|u\|^{l+2}+\|v\|^{l+2}\big).\label{5.8}
\end{aligned}
\end{equation}
As $ l>0$,  choosing $\nu$ so that $$0<\nu< \frac{l}{l+2},$$
and using the upper bound in \eqref{5.3}, we obtain
$$
\mathcal K^{\frac{\nu(l+2)}{l}}(t)\leq\mathcal K(t)
\le \frac{1}{p}\|uv\|_p^p,
$$
and hence
\begin{equation*}
 \epsilon \frac{1}{4k^{\frac{l+2}{l}}}\,
\mathcal K^{\frac{\nu(l+2)}{l}}(t)\big(\|u\|^{l+2}+\|v\|^{l+2}\big)
   \le
\epsilon \frac{1}{4k^{\frac{l+2}{l}}p}\,
\|uv\|_p^p
\big(\|u\|^{l+2}+\|v\|^{l+2}\big).
\end{equation*}
Since $$\|u\|^{l+2}+\|v\|^{l+2}\ge0,$$ it follows immediately that
$$
\epsilon \frac{1}{4k^{\frac{l+2}{l}}}\,
\mathcal K^{\frac{\nu(l+2)}{l}}(t)
\big(\|u\|^{l+2}+\|v\|^{l+2}\big)
\;\le\;
\epsilon \frac{1}{4k^{\frac{l+2}{l}}p}\,\|uv\|_p^p.
$$
Then
\begin{equation}
\begin{aligned}
z'(t)
\geq&\left (1-\nu-\epsilon  \frac{k}{4}\right)
\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)+\epsilon p(1-\alpha) \mathcal{K}(t)+\epsilon\left(\frac{ p(1-\alpha)}{2}+1\right) (\|u_t\|^2+\|v_t\|^2)\label{5.9}\\
& +\epsilon\left(\frac{ p(1-\alpha)}{2}-1 \right) (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2)
+\epsilon \left(p- \epsilon \frac{1}{4k^{\frac{l+2}{l}}p}\right)\|uv\|_p^p.
\end{aligned}
\end{equation}
Choosing $\alpha$ and $k$ such that
\begin{align}
&p(1-\alpha) >2, \nonumber
\\
 \label{k}
k>\left(\frac{\epsilon}{4p^2}\right)^{\frac{l}{l+2}},
\end{align}
and then $\epsilon$ such that
$$
1-\nu-\epsilon  \frac{k}{4}\geq 0,
$$
and we have
$$
z(t)=\mathcal{K}^{1-\nu}(0)+\epsilon \int\limits_{\Omega} [u_{1}u_0+v_{1}v_0]\,dx> 0.$$
We can pick
$$
\nu \in \Big(0, \frac{l}{l+2}\Big), \quad \alpha \in \Big[0, \frac{p-2}{p}\Big), \quad k \in \Big( (\tfrac{\epsilon}{4p^2})^{\frac{l}{l+2}}, \frac{4(1-\nu)}{\epsilon} \Big]
$$
so that we can choose parameters satisfying all conditions simultaneously.
Then \eqref{5.9} becomes
\begin{align}
&z'(t)\ge \epsilon K \left(\mathcal{K}(t)+ \|u_t\|^2+\|v_t\|^2+\|uv\|_p^p\right)\ge 0,\qquad K>0.\label{5.10}
\end{align}
The functional $z(t)$ is increasing on $(0, t_{\mathrm{max}})$  with
$$z(t)\ge z(0)>0\quad \text{for all}\quad t\ge 0.$$
We  estimate $z^{\frac{1}{1-\nu}}(t)$ by using the elementary inequality
$$| \alpha + \beta |^s \le 2^{s-1}( | \alpha |^s+ | \beta |^s) ,\quad s\ge 1,$$
and the Young's inequality
\begin{equation}
\begin{aligned}
&z^{\frac{1}{1-\nu}}(t)\le 2^{\frac{1}{1-\nu}}\left(\mathcal{K}(t)
+\epsilon \|u\|^{\frac{1}{1-\nu}}\|u_t\|^{\frac{1}{1-\nu}}+\epsilon \|v\|^{\frac{1}{1-\nu}}\|v_t\|^{\frac{1}{1-\nu}}\right)
\\
&\le K\left(\mathcal{K}(t)+\|u_t\|^{2}+\|v_t\|^{2}+\|u\|^{\frac{2}{1-2\nu}} +\|v\|^{\frac{2}{1-2\nu}}\right).
\end{aligned}\label{5.11}
\end{equation}
Since $\nu<\frac{1}{2}-\frac{1}{p}$, using the inequality
$$\xi^\tau \le \Big(1 +\frac{1}{s}\Big)(s + \xi),\quad \xi \ge 0,\quad 0 \le\tau \le 1,\quad s > 0,$$
and taking $$\xi=\|u\|_2^p, \qquad \tau=\frac{2}{(1-2\nu)p}<1,$$  and $s=\mathcal{K}(0)$.
We get, for some constants $K_1, K_2>0$,
\begin{equation}
\begin{aligned}
\|u\|^{\frac{2}{1-2\nu}}+\|v\|^{\frac{2}{1-2\nu}} \leq& \left(1 +\frac{1}{\mathcal{K}(t=0)}\right)\left(\mathcal{K}(0) + \|u\|_2^p \right)+ \left(1 +\frac{1}{\mathcal{K}(t=0)}\right)\left(\mathcal{K}(0) + \|v\|_2^p \right) \\
 \leq& K_1\left(\mathcal{K}(t) + \|u\|_p^p+ \|v\|_p^p\right)
 \leq  K_2\left(\mathcal{K}(t) + \sigma\|u v\|_p^p+C(\sigma)
\right), \quad \sigma>0.\label{5.12}
\end{aligned}
\end{equation}
Substituting \eqref{5.12} into \eqref{5.11}, we find
\begin{align}
&z^{\frac{1}{1-\nu}}(t)\le K_3\left(\mathcal{K}(t)+\|u_t\|^{2}+\|v_t\|^{2}+\|uv\|_p^{p}\right), \quad K_3>0.\label{5.13}
\end{align}
By \eqref{5.10}, \eqref{5.13}, we get
\begin{align}
&z'(t)\ge K_4 z^{\frac{1}{1-\nu}}(t), K_4>0.\label{5.14}
\end{align}
A simple integration of \eqref{5.14} over $(0, t)$ then yields
$$z(t)\ge \left(z^{\frac{-\nu}{1-\nu}}(t=0)-K_4\frac{\nu t}{1-\nu}\right)^{\frac{\nu-1}{\nu}}.$$
Thus, there exists $t_2$ such that
\begin{align}
&t_2\le t_*=\frac{z^{\frac{-\nu}{1-\nu}}(t=0)(1-\nu)}{K_4\nu}, \nonumber
\\
&\lim_{t\rightarrow t_2^{-}}z(t)=+\infty.\label{5.15}
\end{align}
The proof is complete.
\end{proof}

\vskip10pt

\subsection{Positive initial energy}
Let
\begin{align}
&0\le E(t=0)\le E_1,\nonumber
\\
&E_{1}
=\left(\frac{1}{2}-\frac{1}{p}\right)C_0^{-\frac{2p}{p-2}}, \label{5.16}
\end{align}
and $C_0$ is the best Poincar\'e  constant.

\begin{lemma}\label{lemma3.5}
    \cite{Wu2009TJM} If $$0\le E(t=0)< E_1, \qquad \|\nabla^\kappa u_0\|+\|\nabla^\kappa v_0\|>\lambda_1,$$
 then there exists $\lambda_2>\lambda_1$ such  that $$\|\nabla^\kappa u\|+\|\nabla^\kappa v\|\ge \lambda_2, \qquad \lambda_1=C_0^{-\frac{p}{p-2}}.$$
\end{lemma}

\begin{theorem} \label{theoremblowup2}Under the assumptions of Theorem~\ref{Theoremlocal}, for  $$0\le E(0)< E_1,\qquad \|\nabla^\kappa u_0\|+\|\nabla^\kappa v_0\|> \lambda_1,$$
 the solution to problem
\eqref{1.1}--\eqref{1.3} blows up, that is, $t_{\mathrm{max}}<+\infty$.
\end{theorem}
\begin{proof}

We set $$\mathcal{K}(t)=E_2-E(t),\qquad E_2=\frac{E(t=0)+E_1}{2}.$$ Proceeding as in \eqref{5.8}  with $\alpha=0$, we obtain
\begin{equation}
\begin{aligned}
z'(t)\ge &\left(1-\nu-\epsilon  \frac{k}{4}\right)
\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)+\epsilon p\mathcal{K}(t)+\epsilon\left(\frac{ p}{2}+1\right) (\|u_t\|^2+\|v_t\|^2)
\\
&+\epsilon\left(\frac{ p}{2}-1\right) (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2)
+\epsilon p\|uv\|_p^p
\\
&- \frac{\epsilon}{4k^{\frac{l+2}{l}}}\mathcal K^{ \frac{\nu(l+2)}{l}}(t) (\|u\|^{l+2}+\|v\|^{l+2}) -\epsilon pE_2.
\end{aligned}\label{5.17}
\end{equation}
Moreover,
\begin{equation}
\begin{aligned}
\epsilon\left(\frac{ p}{2}-1\right) (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2) -\epsilon pE_2
=&\epsilon \left(\left(\frac{ p}{2}-1\right) (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2) -pE_2\right)
\\
=&\epsilon\Big(\left(\frac{ p}{2}-1\right)\frac{\lambda_2^2-\lambda_1^2}{\lambda_2^2} (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2) \\
&+\left(\frac{ p}{2}-1\right)\lambda_1^2\frac{\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2}{\lambda_2^2} -pE_2\Big) \\
\ge &\epsilon \left(K_5(\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2)+K_6\right),
\end{aligned}\label{5.18}
\end{equation}
where $\lambda_2$ is given in Lemma \ref{lemma3.5},
$$K_5=\left(\frac{ p}{2}-1\right)\frac{\lambda_2^2-\lambda_1^2}{\lambda_2^2}>0,\qquad K_6=\left(\frac{ p}{2}-1\right)\lambda_1^2 -pE_2.$$
By \eqref{5.16} and Lemma  \ref{lemma3.5}, we observe that
\begin{equation} K_6=\left(\frac{ p}{2}-1\right)\lambda_1^2 -\frac{p\left(E_1+E(t=0)\right)}{2} =\frac{p\left(E_1-E(t=0)\right)}{2} >0.
 \label{5.19}
\end{equation}
Thus, by \eqref{5.18},\quad \eqref{5.19} and \eqref{5.17}, we arrive at
\begin{equation}
\begin{aligned}
z'(t)\ge& \left(1-\nu-\epsilon  \frac{k}{4}\right)
\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)+\epsilon p\mathcal{K}(t)+\epsilon\left(\frac{ p}{2}+1\right) (\|u_t\|^2+\|v_t\|^2)\nonumber\\
&+\epsilon K_5 (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2)
+\epsilon p\|uv\|_p^p-\epsilon \frac{1}{4k^{\frac{l+2}{l}}}\mathcal K^{ \frac{\nu(l+2)}{l}}(t) (\|u\|^{l+2}+\|v\|^{l+2}).
\end{aligned}\label{5.20}
\end{equation}
Similarly to  \eqref{5.12}  we have
\begin{align*}
z'(t)\ge &\left(1-\nu-\epsilon  \frac{k}{4}\right)
\mathcal{K}^{-\nu}(t)\mathcal{K}'(t)+\epsilon p\mathcal{K}(t)+\epsilon\left(\frac{ p}{2}+1\right) (\|u_t\|^2+\|v_t\|^2)
\\
&+\epsilon K_{5}  (\|\nabla^\kappa u\|^2+\|\nabla^\kappa v\|^2)
+\epsilon \left(p-\epsilon \frac{\sigma}{4k^{\frac{l+2}{l}}p}\right)\|uv\|_p^p+C(\sigma).
\end{align*}
Choosing $k$  as in \eqref{k},  we can complete the proof as in Theorem \ref{theoremblowup}.
\end{proof}

\vskip30pt


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