%\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \documentclass[10pt]{studiamnew} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsfonts} \usepackage{graphicx} \usepackage{amsmath} \usepackage[T1]{fontenc} \usepackage{amssymb} \sloppy \setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{14} \renewcommand{\currentvolume}{} \renewcommand{\currentyear}{2021} \renewcommand{\currentissue}{} \newtheorem{theorem}{Theorem} \newtheorem{example}[theorem]{Example} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} \newcommand{\QATOP}[2]{\genfrac{}{}{0pt}{}{#1}{#2}} \newcommand{\func}[1]{\operatorname{#1}} \newtheorem{notation}[theorem]{Note} \begin{document} \title[On the Petrovesky equation ]{Global existence and blow-up of a Petrovsky equation with general nonlinear dissipative and source terms } \author{Mosbah Kaddour} \address{Faculty of Mathematics and Computer Science,\\ Mohamed Boudiaf University-M'Sila 28000, Algeria} \email{mosbah\_kaddour@yahoo.fr} \author{Farid Messelmi} \address{Department of Mathematics and LDMM Laboratory Ziane Achour, \\ University of Djelfa 17000, Algeria} \email{foudimath@yahoo.fr} \subjclass{Primary:93C20 ; Secondary: 93D15 } \keywords{Global Existence, Blow-up, Nonlinear source, Nonlinear dissipative, Petrovsky Equation} \begin{abstract} This work studies the initial boundary value problem for the Petrovsky equation with nonlinear damping \begin{equation*} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u-\Delta u^{\prime }+\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) =\beta f\left( u\right) \text{ in }\Omega \times \left[ 0,+\infty \right[, \end{equation*} where $\Omega $ is open and bounded domain in $\mathbb{R}^{n}$ with a smooth boundary $\partial \Omega =\Gamma$, $\alpha$, and $\beta >0$. For the nonlinear continuous term $f\left( u\right) $ and for $g$ continuous, increasing, satisfying $g$ $\left( 0\right) $ $=0$, under suitable conditions, the global existence of the solution is proved by using the Faedo-Galerkin argument combined with the stable set method in $% H_{0}^{2}\left( \Omega \right)$. Furthermore, we show that this solution blows up in a finite time when the initial energy is negative. \end{abstract} \maketitle \section{Introduction} This paper devoted to the global existence, uniqueness, and the blow-up of solutions for the nonlinear general Petrovsky equation \begin{equation} \left\{ \begin{array}{c} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u\left( t\right) -\Delta u^{\prime }\left( t\right) +\left\vert u\right\vert ^{p-2}u\left( t\right) +\alpha g\left( u^{\prime }\left( t\right) \right) =\beta f\left( u\left( t\right) \right) ,\text{ in }\Omega \times \mathbb{R}^{+}, \\ u=\partial _{\eta }u=0,\ \text{on}\ \Gamma \times \left[ 0,+\infty \right[ , \\ u(x,0)=u_{0}(x),\text{ }u^{\prime }(x,0)=u_{1}(x)\ \text{in }\Omega .% \end{array}% \right. \label{1} \end{equation}% Recently, in the absence of the strong damping term $-\Delta u^{\prime }\left( t\right) $ and in the case where $\beta f\left( u\left( t\right) \right) =-q\left( x\right) u\left( x,t\right) +\left\vert u\right\vert ^{p-2}u\left( t\right) $ for $g$ continuous, increasing, satisfying $g\left( 0\right) =0$, and $q:\Omega \rightarrow \mathbb{R}^{+},$ a bounded function, the problem\ (\ref{1}) becomes the following \begin{equation*} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u\left( t\right) +q\left( x\right) u\left( x,t\right) +g\left( u^{\prime }\left( t\right) \right) =0,% \text{ in }\Omega \times \mathbb{R}^{+}\text{.} \end{equation*}% This equation together with initial and boundary conditions of Dirichlet type was considered by Guesmia in \cite{Guesmia}, he proved a global existence and a regularity result of the solution, the author under suitable growth conditions on $g$ showed that the solution decays exponentially if $g$ behaves like a linear function, whereas the decay is of a polynomial order otherwise. Without the strong damping term $-\Delta u^{\prime }\left( t\right) $ with $\alpha g\left( u^{\prime }\left( t\right) \right) =\left\vert u^{\prime }\left( t\right) \right\vert ^{\sigma -2}u^{\prime }\left( t\right) $ and $\beta f\left( u\left( t\right) \right) =\left( b+1\right) \left\vert u\left( t\right) \right\vert ^{p-2}u\left( t\right) $, $b>0,$ the problem (\ref{1}) reduced to the following problem \begin{equation*} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u\left( t\right) +\left\vert u^{\prime }\left( t\right) \right\vert ^{\sigma -2}u^{\prime }\left( t\right) =b\left\vert u\left( t\right) \right\vert ^{p-2}u\left( t\right) ,\text{ in }\Omega \times \mathbb{R}^{+}, \end{equation*}% this problem has been considered by Messaoudi in \cite{Messaoudi}, where he investigated the global existence and blow-up of solution. More precisely, he showed that solutions with any initial data continue to exist globally in time if $\sigma \geq p$ and blow-up in finite time if $\sigma 0,$ general Petrovsky problem as in (\ref{1}) becomes% \begin{equation} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u\left( t\right) -\Delta u^{\prime }\left( t\right) +\left\vert u^{\prime }\left( t\right) \right\vert ^{\sigma -1}u^{\prime }\left( t\right) =b\left\vert u\left( t\right) \right\vert ^{p-1}u\left( t\right) \text{,} \label{P1} \end{equation}% this problem was considered by Li et al. \cite{Li}, in \cite{PiskinPolat} and in \cite{NOUR}, the authors obtained global existence, uniform decay of solutions without any interaction between $p$ and $\sigma $, the blow-up of the solution result was established when $\sigma q\geq 1$ and $% a\left( x\right) ,$ $b\left( x\right) >0,$ and $g$ in (\ref{1}) likes $% g\left( u^{\prime }\left( x,t\right) \right) =\alpha \left( x\right) \left\vert u^{\prime }\left( t\right) \right\vert ^{\sigma -2}u^{\prime }\left( t\right) $ with $\sigma \geq 2$ for $\alpha \ :\Omega \rightarrow \mathbb{R}^{+}$ a function, satisfying $\alpha _{1}\geq \alpha \left( x\right) \geq \alpha _{0}>0$. For these purposes, we must establish the global existence of solution for (\ref{1}), we use the variational approach of Faedo--Galerkin approximation combined with the monotonous, compactness, and the stable set method as in \cite{Messaoudi}, \cite{ChenandZhou} and in \cite{PiskinPolat} with some modification in some passages to derive the blow-up result in the infinite time of the solution. \section{Hypotheses} Let us state the precise hypotheses on $p$, $g,$ and $f$. Let $p$ be a positive number with \begin{equation} 2\sigma \geq 2, \\ \left\vert g\left( x\right) \right\vert \leq d_{1}\left\vert x\right\vert +d_{2}\left\vert x\right\vert ^{\sigma -1},\text{ }\forall x\in \mathbb{R},% \text{ }p>\sigma \geq 2,\text{ }d_{i}\geq 0.% \end{array}% \right. \tag{H2} \label{H2} \end{equation}% Let $f\left( x,s\right) \in C^{1}\left( \Omega \times \mathbb{R}\right) $, satisfies:% \begin{equation} sf\left( x,s\right) +k_{1}\left( x\right) \left\vert s\right\vert \geq pF\left( x,s\right) ,\text{ }p>2, \tag{H3} \label{H3} \end{equation}% and the growth conditions% \begin{equation} \left\{ \begin{array}{c} \left\vert f\left( x,s\right) \right\vert \leq l_{1}\left( \left\vert s\right\vert ^{\theta }+k_{2}\left( x\right) \right) , \\ \left\vert f_{s}\left( x,s\right) \right\vert \leq l_{1}\left( \left\vert s\right\vert ^{\theta -1}+k_{3}\left( x\right) \right) \text{ in }\Omega \times \mathbb{R},% \end{array}% \right. \tag{H4} \label{H4} \end{equation}% where $F\left( x,s\right) =\int_{0}^{s}f\left( x,\zeta \right) d\zeta $, with some $l_{0},$ $l_{1}>0$ and the non-negative functions $k_{1}\left( x\right) ,$ $k_{2}\left( x\right) ,$ $k_{3}\left( x\right) \in L^{\infty }\left( \Omega \right) $, a.e. $x\in \Omega ,$ and $1<\theta \leq \frac{% \sigma }{2}<\frac{p}{2}.$ \section{Local Existence} In this section, we establish a local existence result for (\ref{1}) under the assumptions on $f,$ $g,$ and $p$. \begin{theorem} \label{theorem1}Let\ $\left( u_{0},u_{1}\right) \in W\cap L^{p}(\Omega )\times H_{0}^{2}\left( \Omega \right) \cap L^{2\sigma -2}(\Omega )$. Assume that (\ref{H1})-(\ref{H4}) hold. Then problem (\ref{1}) has a unique weak solution $u\left( t\right) $ satisfying: \begin{gather} u\in L^{\infty }(0,T;W\cap L^{p}(\Omega )), \label{6b} \\ u^{\prime }\in L^{\infty }(0,T;H_{0}^{2}\left( \Omega \right) ), \label{6b1} \\ g\left( u^{\prime }\left( t\right) \right) .u^{\prime }\left( t\right) \in L^{1}\left( 0,T;L^{1}(\Omega )\right) , \label{6b2} \\ u^{\prime \prime }\in L^{\infty }(0,T;L^{2}(\Omega )), \label{6b3} \end{gather}% where \begin{equation*} H_{0}^{2}\left( \Omega \right) =\left\{ \varphi \in H^{2}\left( \Omega \right) :\varphi =\partial _{\eta }\varphi =0\text{ on }\partial \Omega \right\} , \end{equation*}% and% \begin{equation*} W=\left\{ \varphi \in H^{4}\left( \Omega \right) \cap H_{0}^{2}\left( \Omega \right) :\Delta \varphi =\partial _{\eta }\Delta \varphi =0\text{ on }% \partial \Omega \right\} . \end{equation*} \end{theorem} Note that throughout this paper, $C$ denotes a generic positive constant depending on $\Omega $ and as all given constants, which may be different from line to line, and is capable of being examined and modified. \begin{proof} We adopt the Galerkin method to construct a global solution. Let $T>0$ be a fixed, and denote by $V_{m}$ the space generated by $\left\{ \varphi _{1},% \text{ }\varphi _{2},...,\text{ }\varphi _{m}\right\} ,$ where the set $% \left\{ \varphi _{m};\text{ }m\in \mathbb{N}\right\} $ is a basis of $% L^{2}(\Omega ),$ $H_{0}^{2}\left( \Omega \right) $, and $H^{4}\left( \Omega \right) \cap H_{0}^{2}\left( \Omega \right) .$ We construct approximate solutions $u_{m}$ $\left( m=1,2,3,...\right) $ in the form% \begin{equation*} u_{m}(t)=\underset{j=1}{\overset{m}{\sum }}K_{jm}(t)w_{j}, \end{equation*}% where $K_{jm}$ are determined by the following ordinary differential equations:% \begin{eqnarray} \left( u_{m}^{\prime \prime },w_{j}\right) +\left( \Delta u_{m},\Delta w_{j}\right) +\left( \nabla u_{m}^{\prime },\nabla w_{j}\right) && \label{6} \\ +\left( \left\vert u_{m}\right\vert ^{p-2}u_{m},w_{j}\right) +\alpha \left( g\left( u_{m}^{\prime }\right) ,w_{j}\right) &=&\beta \left( f\left( u_{m}\right) ,w_{j}\right) , \notag \end{eqnarray}% \begin{eqnarray} u_{m}\left( 0\right) &=&u_{0m}=\sum_{i=1}^{m}\left( u_{0},w_{j}\right) w_{j}% \overset{\text{as\ }m\longrightarrow \infty }{\longrightarrow }u_{0}\ \label{7} \\ &&\text{in }H^{4}\left( \Omega \right) \cap H_{0}^{2}\left( \Omega \right) \cap L^{p}(\Omega )\text{ }, \notag \end{eqnarray}% \begin{eqnarray} u_{m}^{\prime }\left( 0\right) &=&u_{1m}=\sum\limits_{i=1}^{m}\left( u_{1},w_{j}\right) w_{j}\overset{\text{as\ }m\longrightarrow \infty }{% \longrightarrow }u_{1}\ \label{8} \\ &&\text{in }H_{0}^{2}\left( \Omega \right) \cap L^{2\sigma -2}(\Omega ), \notag \end{eqnarray}% with $u_{0},$ $u_{1}$ are given functions on $\Omega $, by virtue of the theory of ordinary differential equations, the system (\ref{6})-(\ref{8}) has a unique local solution on some interval $\left[ 0,t_{m}\right) $. We claim that for any $T>0$, such a solution can be extended to the whole interval $\left[ 0,T\right] $, as a consequence of the a priori estimates that shall be proven in the next step. We denote by $C,$ $C_{k}$ or $c_{k}$ the constants which are independent of $m$, the initial data $u_{0}$ and $% u_{1}$. Multiplying the equation (\ref{6}) by $K_{jm}^{\prime }(t)$ and performing the summation over $j=1,...,m$, the integration par parts gives\qquad \begin{equation} E_{m}^{\prime }\left( t\right) +\left\vert \nabla u_{m}^{\prime }\left( t\right) \right\vert ^{2}+\alpha \left( g\left( u_{m}^{\prime }\left( t\right) \right) ,u_{m}^{\prime }\left( t\right) \right) =0,\text{ }\forall t\geq 0\text{,} \label{9} \end{equation}% where \begin{equation} E_{m}\left( t\right) =\frac{1}{2}\left\vert u_{m}^{\prime }(t)\right\vert ^{2}+\frac{1}{2}\left\vert \Delta u_{m}(t)\right\vert ^{2}+\frac{1}{p}% \left\Vert u_{m}(t)\right\Vert _{p}^{p}-\beta \int_{\Omega }F\left( x,u_{m}(t)\right) dx, \label{10} \end{equation}% by (\ref{H3}), and Young\ inequality, we have% \begin{gather} -\int_{\Omega }F\left( x,u_{m}\right) dx\geq -\frac{1}{p}\int_{\Omega }k_{1}\left( x\right) \left\vert u_{m}\right\vert dx-\frac{1}{p}\int_{\Omega }u_{m}f\left( x,u_{m}\right) dx \label{11} \\ \geq -\varepsilon C_{\ast }^{2}\left\vert \Delta u_{m}(t)\right\vert ^{2}-C_{\varepsilon }\left\vert k_{1}\left( x\right) \right\vert ^{2}-\frac{1% }{p}\int_{\Omega }u_{m}f\left( x,u_{m}\right) dx, \notag \end{gather}% by using hypotheses (\ref{H4}), Young's inequality yields \begin{gather} \frac{1}{p}\int_{\Omega }u_{m}f\left( x,u_{m}\right) dx\leq \frac{1}{p}% \left\vert f\left( x,u_{m}\right) \right\vert \left\vert u_{m}\right\vert \notag \\ \leq \frac{l_{1}^{2}}{p}\varepsilon \int_{\Omega }\left( \left\vert u_{m}\right\vert ^{2\theta }+\left\vert k_{2}\left( x\right) \right\vert ^{2}\right) dx+\frac{c\left( \varepsilon ,p\right) }{p^{2}}\int_{\Omega }\left\vert u_{m}\right\vert ^{2}dx \notag \\ =\frac{l_{1}^{2}}{p}\varepsilon \left\Vert u_{m}\right\Vert _{2\theta }^{2\theta }+\frac{l_{1}^{2}}{p}\varepsilon \left\vert k_{2}\left( x\right) \right\vert ^{2}+\frac{c\left( \varepsilon ,p\right) }{p^{2}}\left\Vert u_{m}\right\Vert _{p}^{2} \label{12} \\ \leq \frac{l_{1}^{2}}{p}\varepsilon \left( \frac{p-2\theta }{p}+\frac{% 2\theta }{p}\left\Vert u_{m}\right\Vert _{p}^{p}\right) +\frac{l_{1}^{2}}{p}% \varepsilon \left\vert k_{2}\left( x\right) \right\vert ^{2} \notag \\ +C^{\prime }\left( \varepsilon ,p\right) +\frac{\text{ }1}{p^{2}}\left\Vert u_{m}\right\Vert _{p}^{p}, \notag \end{gather}% substituting (\ref{12}) in (\ref{11}), and chosen $\varepsilon \leq C_{0}=\min \left( \frac{1}{2C_{\ast }^{2}};\frac{p}{2\theta l_{1}^{2}+1}% \right) $, (\ref{10}) becomes \begin{equation} E_{m}\left( t\right) \geq \frac{1}{2}\left\vert u_{m}^{\prime }(t)\right\vert ^{2}+C_{1}\left\vert \Delta u_{m}(t)\right\vert ^{2}+C_{2}\left\Vert u_{m}\right\Vert _{p}^{p}-C_{3}\left( 1+K_{1}+K_{2}\right) , \label{13} \end{equation}% or% \begin{equation} \left\vert u_{m}^{\prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}(t)\right\vert ^{2}+\left\Vert u_{m}\right\Vert _{p}^{p}\leq C_{4}\left( E_{m}\left( t\right) +K_{1}+K_{2}+1\right) , \label{14} \end{equation}% where \begin{gather*} 00$, where $K_{1}=\left\Vert k_{1}\right\Vert _{\infty }^{2},$ $% K_{2}=\left\Vert k_{2}\right\Vert _{\infty }^{2}.$ Hence, for any $t\geq 0$, and $m=1,2,...,$ from (\ref{15}), and (\ref{17}) we get \begin{equation} \left\vert u_{m}^{\prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}(t)\right\vert ^{2}+\int_{0}^{t}\left\vert \nabla u_{m}^{\prime }\left( s\right) \right\vert ^{2}ds+\left\Vert u_{m}\left( t\right) \right\Vert _{p}^{p}+\alpha \int_{0}^{t}\int_{\Omega }g\left( u_{m}^{\prime }\left( s\right) \right) u_{m}^{\prime }\left( s\right) dxds\leq C. \label{18} \end{equation}% By the growth conditions, the estimate (\ref{18}), and as $2\theta \leq p,$ we have \begin{equation*} \left\vert f\left( u_{m}\right) \right\vert ^{2}\leq Cl_{1}\left( \left\vert u_{m}\right\vert ^{2\theta }+\left\vert k_{2}\left( x\right) \right\vert ^{2}\right) \leq C\left( \left\vert \left\vert u_{m}\right\vert \right\vert _{p}^{2\theta }+\left\Vert k_{2}\right\Vert _{\infty }^{2}\right) \leq C. \end{equation*}% With this estimate we can extend the approximate solution $u_{m}\left( t\right) $ to the interval $\left[ 0,T\right] $ and the following a priori estimates% \begin{equation} \left\{ \begin{array}{l} u_{m}\text{ is bounded in}\ L^{\infty }\left( 0,T;L^{p}(\Omega )\right) , \\ u_{m}^{\prime }\text{ is bounded in}\ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) , \\ \nabla u_{m}^{\prime }\text{ is bounded in}\ L^{2}\left( 0,T;L^{2}(\Omega )\right) , \\ g\left( u_{m}^{\prime }\right) .u_{m}^{\prime }\text{ is bounded in}\ L^{1}\left( \Omega \times \left( 0,T\right) \right) , \\ \Delta u_{m}(t)\text{ is bounded in}\ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) , \\ f\left( u_{m}\right) \text{ is bounded in}\ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) ,% \end{array}% \right. \label{19} \end{equation}% hold. \end{proof} \begin{lemma} There exists a constant $K>0$ such that% \begin{equation*} \left\vert \left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \right\vert _{L^{\frac{\sigma }{\sigma -1}}\left( \Omega \times % \left[ 0,T\right] \right) }\leq K, \end{equation*}% for all $m\in\mathbb{N}.$ \end{lemma} \begin{proof} From (\ref{H2}), Holder's, and Young's inequalities gives% \begin{gather*} \int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\right) \right\vert ^{\frac{\sigma }{\sigma -1}}dxdt=\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\right) \right\vert \left\vert g\left( u_{m}^{\prime }\right) \right\vert ^{\frac{1}{\sigma -1}}dxdt \\ \leq \int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left( d_{1}\left\vert u_{m}^{\prime }\left( t\right) \right\vert +d_{2}\left\vert u_{m}^{\prime }\left( t\right) \right\vert ^{\sigma -1}\right) ^{\frac{1}{\sigma -1}}dxdt \\ \leq C\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left( \left\vert u_{m}^{\prime }\left( t\right) \right\vert ^{\frac{1}{\sigma -1}}+\left\vert u_{m}^{\prime }\left( t\right) \right\vert \right) dxdt \\ =C\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left\vert u_{m}^{\prime }\left( t\right) \right\vert ^{\frac{1}{\sigma -1}}dxdt \\ +C\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left\vert u_{m}^{\prime }\left( t\right) \right\vert dxdt \\ \leq \frac{\sigma -1}{\sigma }\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\right) \right\vert ^{\frac{\sigma }{\sigma -1}}dxdt+C\left( \sigma \right) \int_{0}^{T}\int_{\Omega }\left\vert u_{m}^{\prime }\left( t\right) \right\vert ^{\frac{\sigma }{\sigma -1}}dxdt \\ +C\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left\vert u_{m}^{\prime }\left( t\right) \right\vert dxdt, \end{gather*}% therefore% \begin{gather*} \frac{1}{\sigma }\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert ^{\frac{\sigma }{\sigma -1}}dxdt\leq C\left( \sigma \right) \int_{0}^{T}\int_{\Omega }\left\vert u_{m}^{\prime }\left( t\right) \right\vert ^{\frac{\sigma }{\sigma -1}}dxdt \\ +C\int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left\vert u_{m}^{\prime }\left( t\right) \right\vert dxdt \\ \leq C\int_{0}^{T}\left\vert \left\vert u_{m}^{\prime }\left( t\right) \right\vert \right\vert _{2}^{\frac{\sigma }{\sigma -1}}dt+C\int_{0}^{T}% \int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert \left\vert u_{m}^{\prime }\left( t\right) \right\vert dxdt, \end{gather*}% hence, by (\ref{19}), we deduce \begin{equation*} \int_{0}^{T}\int_{\Omega }\left\vert g\left( u_{m}^{\prime }\left( t\right) \right) \right\vert ^{\frac{\sigma }{\sigma -1}}dxdt\leq K. \end{equation*} \end{proof} \begin{lemma} There exists a constant $M>0$ such that% \begin{equation*} \left\vert u_{m}^{\prime \prime }(t)\right\vert +\left\vert \Delta u_{m}^{\prime }(t)\right\vert +\int_{0}^{T}\left\vert \nabla u_{m}^{\prime \prime }\left( t\right) \right\vert dt\leq M, \end{equation*}% for all $m\in\mathbb{N}.$ \end{lemma} \begin{proof} From (\ref{6}) we obtain% \begin{equation*} \left\vert u_{m}^{\prime \prime }(0)\right\vert \leq \left\vert u_{0m}\right\vert ^{p-1}+\left\vert \Delta ^{2}u_{0m}\right\vert +\left\vert \Delta u_{1m}\right\vert +\alpha \left\vert g\left( u_{1m}\right) \right\vert +\beta \left\vert f\left( u_{0m}\right) \right\vert , \end{equation*}% by (\ref{H4}) we have \begin{equation*} \left\vert f\left( u_{0m}\right) \right\vert ^{2}\leq l_{1}\left( \left\vert u_{0m}\right\vert ^{2\theta }+\left\vert k_{2}\left( x\right) \right\vert ^{2}\right) \leq C\left( \left\vert \left\vert \Delta u_{0m}\right\vert \right\vert _{2}^{2\theta }+\left\Vert k_{2}\right\Vert _{\infty }^{2}\right) , \end{equation*}% Since $g\left( u_{1m}\right) $ is bounded in $L^{2}\left( \Omega \right) $ by (\ref{H2}), from (\ref{7}) and (\ref{8}) we obtain% \begin{equation*} \left\vert u_{m}^{\prime \prime }(0)\right\vert \leq C. \end{equation*}% Differentiating (\ref{6}) with respect to $t$, we get% \begin{gather} \left( u_{m}^{\prime \prime \prime },w_{j}\right) +\left( \Delta ^{2}u_{m}^{\prime },w_{j}\right) -\left( \Delta u_{m}^{\prime \prime },w_{j}\right) +\left( p-1\right) \left( \left\vert u_{m}\right\vert ^{p-2}u_{m}^{\prime },w_{j}\right) \notag \\ +\alpha \left( g^{\prime }\left( u_{m}^{\prime }\right) u_{m}^{\prime \prime },w_{j}\right) =\beta \left( f^{\prime }\left( u_{m}\right) u_{m}^{\prime },w_{j}\right) . \label{20} \end{gather}% Multiplying it by $K_{jm}^{\prime \prime }(t)$ and summing over $j$ from $1$ to $m$ , according to the H\"{o}lder's inequality, to find% \begin{gather} \frac{1}{2}\frac{d}{dt}\left( \left\vert u_{m}^{\prime \prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}^{\prime }\left( t\right) \right\vert ^{2}\right) +\left\vert \nabla u_{m}^{\prime \prime }\left( t\right) \right\vert ^{2}+\alpha \left( g^{\prime }\left( u_{m}^{\prime }\right) u_{m}^{\prime \prime },u_{m}^{\prime \prime }\right) \label{21} \\ \leq \left( p-1\right) \int_{\Omega }\left\vert u_{m}\right\vert ^{p-2}\left\vert u_{m}^{\prime }\right\vert \left\vert u_{m}^{\prime \prime }\right\vert dx+\beta \int_{\Omega }\left\vert f^{\prime }\left( u_{m}\right) \right\vert \left\vert u_{m}^{\prime }\right\vert \left\vert u_{m}^{\prime \prime }\right\vert dx. \notag \end{gather}% By choosing $\lambda $ satisfies the inequalities \begin{equation*} \left\{ \begin{array}{l} \lambda +1\leq \min \left( \frac{p}{2\left( \theta -1\right) },\frac{n}{n-4}% \right) \text{ if }n\geq 5, \\ \lambda +1\leq \frac{p}{2\left( \theta -1\right) }\text{ if }n=1,2,3,4,% \end{array}% \right. \end{equation*}% then by using (\ref{H4}), estimates (\ref{19}) and generalized H\"{o}lder's inequality, we deduce that \begin{gather} \int_{\Omega }\left\vert f^{\prime }\left( u_{m}\right) \right\vert \left\vert u_{m}^{\prime }\right\vert \left\vert u_{m}^{\prime \prime }\right\vert dx \notag \\ \leq \left\Vert l_{1}\left( \left\vert u_{m}\right\vert ^{\theta -1}+k_{3}\left( x\right) \right) \right\Vert _{2\left( \lambda +1\right) }^{\lambda }\left\vert \left\vert u_{m}^{\prime }\right\vert \right\vert _{2\left( \lambda +1\right) }\left\Vert u_{m}^{\prime \prime }\right\Vert _{2} \notag \\ \leq C\left( \left\Vert \left\vert u_{m}\right\vert ^{\theta -1}\right\Vert _{2\left( \lambda +1\right) }^{\lambda }+\left\Vert k_{3}\left( x\right) \right\Vert _{2\left( \lambda +1\right) }^{\lambda }\right) \left\vert \left\vert u_{m}^{\prime }\right\vert \right\vert _{2\left( \lambda +1\right) }\left\Vert u_{m}^{\prime \prime }\right\Vert _{2} \notag \\ \leq C\left( \left\Vert u_{m}\right\Vert _{p}^{\lambda \left( \theta -1\right) }+\left\Vert k_{3}\left( x\right) \right\Vert _{p}^{\lambda }\right) \left\vert \left\vert \Delta u_{m}^{\prime }\right\vert \right\vert _{2}\left\Vert u_{m}^{\prime \prime }\right\Vert _{2} \notag \\ \leq C_{5}\left( \left\vert u_{m}^{\prime \prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}^{\prime }\left( t\right) \right\vert ^{2}\right) , \label{22} \end{gather}% where $C_{1}$ and $C_{2}$ are positive constants independent of $m$ and $% t\in \left[ 0,T\right] .$ By same manner, using condition (\ref{H1}), Young's inequality, Sobolev embedding, and estimate (\ref{19}) we reach to \begin{gather} \int_{\Omega }\left\vert u_{m}\right\vert ^{p-2}\left\vert u_{m}^{\prime }\right\vert \left\vert u_{m}^{\prime \prime }\right\vert dx\leq \left\Vert \left\vert u_{m}\right\vert ^{p-2}\right\Vert _{n}\left\Vert u_{m}^{\prime }\right\Vert _{\frac{2n}{n-2}}\left\Vert u_{m}^{\prime \prime }\right\Vert _{2} \notag \\ \leq C\left\vert \left\vert \Delta u_{m}^{\prime }\right\vert \right\vert _{2}\left\Vert u_{m}^{\prime \prime }\right\Vert _{2}\leq C_{5}\left( \left\vert u_{m}^{\prime \prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}^{\prime }\left( t\right) \right\vert ^{2}\right) . \label{22b} \end{gather}% Combining (\ref{21}), (\ref{22}) and (\ref{22b}) we deduce \begin{gather*} \frac{1}{2}\frac{d}{dt}\left( \left\vert u_{m}^{\prime \prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}^{\prime }\left( t\right) \right\vert ^{2}\right) +\left\vert \nabla u_{m}^{\prime \prime }\left( t\right) \right\vert ^{2}+\alpha \left( g^{\prime }\left( u_{m}^{\prime }\right) u_{m}^{\prime \prime },u_{m}^{\prime \prime }\right) \\ \leq C_{6}\left( \left\vert u_{m}^{\prime \prime }(t)\right\vert ^{2}+\left\vert \Delta u_{m}^{\prime }\left( t\right) \right\vert ^{2}\right) . \end{gather*}% Integrating the last inequality over $(0,t)$ and applying Gronwall's lemma, we obtain% \begin{equation*} \left\vert u_{m}^{\prime \prime }(t)\right\vert +\left\vert \Delta u_{m}^{\prime }\left( t\right) \right\vert +\int_{0}^{t}\left\vert \nabla u_{m}^{\prime \prime }\left( t\right) \right\vert ^{2}ds\leq C\text{ for all }t\geq 0. \end{equation*}% Therefore% \begin{gather} u_{m}^{\prime \prime }\text{ is bounded in}\ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) , \notag \\ \Delta u_{m}^{\prime }\text{ is bounded in}\ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) , \label{23} \\ \nabla u_{m}^{\prime \prime }\text{ is bounded in}\ L^{2}\left( 0,T;L^{2}(\Omega )\right) , \notag \end{gather}% it follows from (\ref{23}), $\left( u_{m}^{\prime }\right) $ is bounded in$\ L^{\infty }\left( 0,T;H_{0}^{2}(\Omega )\right) .$ Furthermore, by applying the Lions-Aubin compactness Lemma in \cite{Lions2}, we claim that \begin{equation} u_{m}^{\prime }\text{ is compact in}\ L^{2}\left( 0,T;L^{2}(\Omega )\right) , \label{24} \end{equation} From (\ref{19}) and (\ref{23}), there exists a subsequence of $\left( u_{m}\right) $, still denote by $\left( u_{m}\right) $, such that% \begin{equation} \left\{ \begin{array}{c} u_{m}\longrightarrow u\ \text{weak star in}\ L^{\infty }\left( 0,T;H_{0}^{2}(\Omega )\right) \text{,} \\ u_{m}\longrightarrow u\ \text{strongly in}\ L^{2}\left( 0,T;L^{2}(\Omega )\right) \text{,} \\ u_{m}^{\prime }\longrightarrow u^{\prime }\ \text{weak star in}\ L^{\infty }\left( 0,T;H_{0}^{2}(\Omega )\right) \text{,} \\ u_{m}^{\prime }\longrightarrow u^{\prime }\ \text{strongly in}\ L^{2}\left( 0,T;L^{2}(\Omega )\right) \text{,} \\ u_{m}^{\prime \prime }\longrightarrow u^{\prime \prime }\ \text{weak star in}% \ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) \text{,} \\ g\left( u_{m}^{\prime }\right) \longrightarrow \chi \ \text{weak star in}\ L^{\frac{\sigma }{\sigma -1}}\left( \Omega \times \left( 0,T\right) \right) \text{,} \\ f\left( u_{m}\right) \longrightarrow \zeta \ \text{weak star in}\ L^{\infty }\left( 0,T;L^{2}(\Omega )\right) \text{.}% \end{array}% \right. \label{27} \end{equation}% Using the compactness of $H_{0}^{2}(\Omega )$ to $L^{2}(\Omega )$, it is easy to see that% \begin{equation*} \int_{0}^{T}\int_{\Omega }\left\vert u_{m}\right\vert ^{p-2}u_{m}vdxdt\rightarrow \int_{0}^{T}\int_{\Omega }\left\vert u\right\vert ^{p-2}uvdxdt,\text{ for all }v\in L^{\sigma }\left( 0,T;H_{0}^{2}(\Omega )\right) , \end{equation*}% as $m\rightarrow \infty .$ By (\ref{H2}), and estimates (\ref{27}) we have \begin{equation*} g\left( u_{m}^{\prime }\right) \longrightarrow g\left( u^{\prime }\right) \text{ a.e.in }\Omega \times \left( 0,T\right) . \end{equation*}% Therefore, from \cite[Chapter1,Lemma1.3]{Lions2}, we infer that \begin{equation*} g\left( u_{m}^{\prime }\right) \longrightarrow g\left( u^{\prime }\right) \ \text{weak star in}\ L^{\frac{\sigma }{\sigma -1}}\left( 0,T;L^{\frac{\sigma }{\sigma -1}}\right) \text{,} \end{equation*}% as $m\rightarrow \infty ,$ and this implies that% \begin{equation*} \int_{0}^{T}\int_{\Omega }g\left( u_{m}^{\prime }\right) vdxdt\rightarrow \int_{0}^{T}\int_{\Omega }g\left( u^{\prime }\right) vdxdt\ \text{\ for all }% v\in L^{\sigma }\left( 0,T;H_{0}^{2}(\Omega )\right) . \end{equation*}% By the same manner using the growth conditions in (\ref{H4}) and estimate (% \ref{27}), we see that% \begin{equation*} \int_{0}^{T}\int_{\Omega }\left\vert f\left( u_{m}\right) \right\vert ^{% \frac{\theta +1}{\theta }}dxdt \end{equation*}% is bounded and \begin{equation*} f\left( u_{m}\right) \longrightarrow f\left( u\right) \text{ a.e.in }\Omega \times \left( 0,T\right) , \end{equation*}% then \begin{equation*} f\left( u_{m}\right) \longrightarrow f\left( u\right) \ \text{weak star in}\ L^{\frac{\theta +1}{\theta }}\left( 0,T;L^{\frac{\theta +1}{\theta }}\right) \text{,} \end{equation*}% as $m\rightarrow \infty ,$ and this implies that \begin{equation*} \int_{0}^{T}\int_{\Omega }f\left( u_{m}\right) vdxdt\rightarrow \int_{0}^{T}\int_{\Omega }f\left( u\right) vdxdt\ \text{\ for all }v\in L^{\theta }\left( 0,T;H_{0}^{2}(\Omega )\right) . \end{equation*}% It follows at once from all estimates that for each fixed $v\in L^{\theta }\left( 0,T;H_{0}^{2}(\Omega )\right) \cap L^{\sigma }\left( 0,T;H_{0}^{2}(\Omega )\right) ,$ \begin{gather*} \int_{0}^{T}\int_{\Omega }\left( u_{m}^{\prime \prime }+\Delta ^{2}u_{m}-\Delta u_{m}^{\prime }+\left\vert u_{m}\right\vert ^{\rho }u_{m}+\alpha g\left( u_{m}^{\prime }\right) -\beta f\left( u_{m}\right) \right) vdxdt \\ \rightarrow \int_{0}^{T}\int_{\Omega }\left( u^{\prime \prime }+\Delta ^{2}u-\Delta u^{\prime }+\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) -\beta f\left( u\right) \right) vdxdt, \end{gather*}% as $m\rightarrow \infty .$ Consequently \begin{gather*} \int_{0}^{T}\int_{\Omega }\left( u^{\prime \prime }+\Delta ^{2}u-\Delta u^{\prime }+\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) -\beta f\left( u\right) \right) vdxdt=0, \\ \forall v\in L^{\theta }\left( 0,T;H_{0}^{2}(\Omega )\right) \cap L^{\sigma }\left( 0,T;H_{0}^{2}(\Omega )\right) . \end{gather*}% This means that the problem admit a weak solution $u$ satisfying (\ref{1}), and (\ref{6b})-(\ref{6b3}). \end{proof} \begin{theorem} \label{theorem2}Under the hypotheses of the Theorem \ref{theorem1}, we have the solution $u$ given by Theorem \ref{theorem1}, is unique. \end{theorem} \begin{proof} Let $u$\ and $v$\ are two solutions, in the sense of the Theorem \ref% {theorem1}. Then $w=u-v$\ satisfies% \begin{gather} w^{\prime \prime }+\left( \Delta ^{2}u-\Delta ^{2}v\right) -\Delta w^{\prime }+\alpha \left( g\left( u^{\prime }\right) -g\left( v^{\prime }\right) \right) \notag \\ +(\left\vert u\right\vert ^{p-2}u-\left\vert v\right\vert ^{p-2}v)=\beta \left( f\left( u\right) -f\left( v\right) \right) , \label{28} \end{gather}% \begin{equation} w(0)=w^{\prime }(0)=0\ \text{in}\ \Omega , \label{29} \end{equation}% \begin{equation} w=\ \partial _{\eta }w=0\ \text{on}\ \Sigma , \label{30} \end{equation}% \begin{equation} w\in L^{p}(0,T;W\cap L^{p}(\Omega )), \label{31} \end{equation}% \begin{equation} w^{\prime }\in L^{2}(0,T;H_{0}^{2}\left( \Omega \right) ). \label{32} \end{equation}% Let's multiply the two members of (\ref{28}) by $w^{\prime }$ and integrate on $\Omega $. According to the Green's formula and conditions (\ref{30}), integrating by part the result on $[0,t]$, using conditions (\ref{29}) to find that% \begin{gather} \frac{1}{2}\left( \left\vert w^{\prime }(t)\right\vert ^{2}+\left\vert \Delta w\right\vert ^{2}\right) \leq \int_{0}^{t}\int_{\Omega }\left\vert \left\vert u\right\vert ^{p-2}u-\left\vert v\right\vert ^{p-2}v\right\vert \left\vert w^{\prime }\right\vert dxds \label{33} \\ +\beta \int_{0}^{t}\int_{\Omega }\left\vert f\left( u\right) -f\left( v\right) \right\vert \left\vert w^{\prime }\right\vert dxds. \notag \end{gather}% According to the H\"{o}lder's, Young's inequalities, condition (\ref{H1}), the estimates (\ref{27}) the first term on the right-hand side of (\ref{33}) can be estimated as follows:% \begin{gather} \int_{0}^{t}\int_{\Omega }\left\vert \left\vert u\right\vert ^{p-2}u-\left\vert v\right\vert ^{p-2}v\right\vert \left\vert w^{\prime }\right\vert dxds \notag \\ \leq \left( p-1\right) \int_{0}^{t}\left( \left\Vert \left\vert u\right\vert ^{p-2}\right\Vert _{L^{n}(\Omega )}+\left\Vert \left\vert v\right\vert ^{p-2}\right\Vert _{L^{n}(\Omega )}\right) \left\Vert w\right\Vert _{L^{% \frac{2n}{n-2}}(\Omega )}\left\Vert w^{\prime }\right\Vert _{L^{2}(\Omega )}ds \notag \\ \leq C\int_{0}^{t}\left( \left\Vert u\right\Vert _{L^{n\left( p-2\right) }(\Omega )}^{p-2}+\left\Vert v\right\Vert _{L^{n\left( p-2\right) }(\Omega )}^{p-2}\right) \left\Vert \Delta w\right\Vert _{L^{2}(\Omega )}\left\Vert w^{\prime }\right\Vert _{L^{2}(\Omega )}ds \label{34} \\ \leq C\int_{0}^{t}\left( \left\Vert \Delta u\right\Vert _{L^{2}\left( \Omega \right) }^{p-2}+\left\Vert \Delta v\right\Vert _{L^{2}(\Omega )}^{p-2}\right) \left\Vert \Delta w\right\Vert _{L^{2}(\Omega )}\left\Vert w^{\prime }\right\Vert _{L^{2}(\Omega )}ds \notag \\ \leq C\int_{0}^{t}\left( \left\vert w^{\prime }(s)\right\vert ^{2}+\left\vert \Delta w\left( s\right) \right\vert ^{2}\right) ds. \notag \end{gather}% Now let $U_{\varepsilon }=\varepsilon u+\left( 1-\varepsilon \right) v,$ $% 0\leq \varepsilon \leq 1$, by the growth conditions, for the second term of the right side to (\ref{33}), we have \begin{gather*} \left\vert \int_{0}^{t}\int_{\Omega }\left\vert f\left( u\right) -f\left( v\right) \right\vert \left\vert w^{\prime }\right\vert dxdt\right\vert =\left\vert \int_{0}^{t}\int_{\Omega }\int_{0}^{1}\frac{d}{d\varepsilon }% f\left( U_{\varepsilon }\right) d\varepsilon w^{\prime }dxds\right\vert \\ \leq \int_{0}^{t}\int_{\Omega }\left\vert \int_{0}^{1}\frac{d}{d\varepsilon }% f\left( U_{\varepsilon }\right) d\varepsilon \right\vert \left\vert w^{\prime }\right\vert dxds \\ \leq \int_{0}^{t}\int_{\Omega }\int_{0}^{1}\left\vert \frac{d}{d\varepsilon }% f\left( U_{\varepsilon }\right) d\varepsilon \right\vert \left\vert w^{\prime }\right\vert dxds \\ \leq l_{1}\int_{0}^{t}\int_{\Omega }\int_{0}^{1}\left( \left\vert U_{\varepsilon }\right\vert ^{\theta -1}+\left\vert k_{3}\left( x\right) \right\vert \right) \left\vert u-v\right\vert \left\vert w^{\prime }\right\vert d\varepsilon dxds \\ \leq C\int_{0}^{t}\int_{\Omega }\left( \left\vert u\right\vert ^{\theta -1}+\left\vert v\right\vert ^{\theta -1}+\left\vert k_{3}\left( x\right) \right\vert \right) \left\vert w\left( s\right) \right\vert \left\vert w^{\prime }\left( s\right) \right\vert dxds=I. \end{gather*}% Using\ the generalized H\"{o}lder's, Young's inequalities, and the estimates (\ref{27}), and choosing $\lambda $ such taht% \begin{equation*} \left\{ \begin{array}{l} \lambda +1\leq \frac{n}{\left( \theta -1\right) \left( n-4\right) }\text{ if }n\geq 5, \\ 2\leq \lambda +1<\infty \text{ if }n=1,2,3,4,% \end{array}% \right. \end{equation*}% we infer% \begin{gather} I\leq C\int_{0}^{t}\left\Vert \left\vert u\right\vert ^{\theta -1}+\left\vert v\right\vert ^{\theta -1}+\left\vert k_{3}\left( x\right) \right\vert \right\Vert _{2\left( \lambda +1\right) }^{\lambda }\left\vert \left\vert w\right\vert \right\vert _{2\left( \lambda +1\right) }\left\Vert w^{\prime }\right\Vert _{2} \notag \\ \leq C\int_{0}^{t}\left( \left\Vert \left\vert u\right\vert ^{\theta -1}\right\Vert _{2\left( \lambda +1\right) }^{\lambda }+\left\Vert \left\vert v\right\vert ^{\theta -1}\right\Vert _{2\left( \lambda +1\right) }^{\lambda }+\left\Vert k_{3}\left( x\right) \right\Vert _{2\left( \lambda +1\right) }^{\lambda }\right) \left\vert \left\vert w\right\vert \right\vert _{2\left( \lambda +1\right) }\left\Vert w^{\prime }\right\Vert _{2}ds \notag \\ \leq C\int_{0}^{t}\left( \left\Vert \Delta u\right\Vert _{2}^{\lambda \left( \theta -1\right) }+\left\Vert \Delta v\right\Vert _{2}^{\lambda \left( \theta -1\right) }+\left\Vert k_{3}\left( x\right) \right\Vert _{\infty }^{\lambda }\right) \left\vert \left\vert \Delta w\right\vert \right\vert _{2}\left\Vert w^{\prime }\right\Vert _{2}ds \notag \\ \leq C\int_{0}^{t}\left\vert \left\vert \Delta w\right\vert \right\vert _{2}\left\Vert w^{\prime }\right\Vert _{2}ds\leq C\int_{0}^{t}\left( \left\vert w^{\prime }(s)\right\vert ^{2}+\left\vert \Delta w\left( s\right) \right\vert ^{2}\right) ds. \label{35} \end{gather}% Combining (\ref{33}), (\ref{34}) and (\ref{35}) to obtain% \begin{equation*} \left\vert w^{\prime }(t)\right\vert ^{2}+\left\vert \Delta w\left( t\right) \right\vert ^{2}\leq C\int_{0}^{t}\left( \left\vert w^{\prime }(s)\right\vert ^{2}+\left\vert \Delta w\left( s\right) \right\vert ^{2}\right) ds. \end{equation*}% The integral inequality and Gronwall's lemma show that $w=0.$ \end{proof} \section{Global existence} In this section, we discuss the global existence of the solution for problem (\ref{1}). In order to state and prove our main results, we first introduce the following functions% \begin{equation} I\left( t\right) =I\left( u\left( t\right) \right) =\left\vert \Delta u\left( t\right) \right\vert ^{2}-\beta \int_{\Omega }f\left( u\left( t\right) \right) u\left( x,t\right) dx-\beta \int\limits_{\Omega }k_{1}\left( x\right) \left\vert u\left( x,t\right) \right\vert dx, \label{2g} \end{equation}% \begin{equation} J\left( t\right) =J\left( u\left( t\right) \right) =\frac{1}{2}\left\vert \Delta u\right\vert ^{2}-\beta \int_{\Omega }F\left( x,u\right) dx, \label{3g} \end{equation}% \begin{equation} E(t)=E(u(t),u^{\prime }(t))=J\left( u\left( t\right) \right) +\frac{1}{2}% \left\vert u_{t}(t)\right\vert _{2}^{2}+\frac{1}{p}\left\Vert u(t)\right\Vert _{p}^{p}. \label{4g} \end{equation}% And the stable set as% \begin{equation} W=\left\{ u:u\in H_{0}^{2}\left( \Omega \right) \text{, }I\left( t\right) >0\right\} \cup \left\{ 0\right\} . \label{5g} \end{equation} The next lemma shows that our energy functional (\ref{4g}) is a nonincreasing function along with the solution of (\ref{1}). \begin{lemma} \label{lemma1}$E(t)$ is a nonincreasing function for $t\geq 0$ and% \begin{equation} E^{\prime }\left( t\right) =-\left\vert \nabla u^{\prime }\left( t\right) \right\vert ^{2}-\alpha \int_{\Omega }u^{\prime }\left( t\right) g\left( u^{\prime }\left( t\right) \right) dx\leq 0. \label{8g} \end{equation} \end{lemma} \begin{proof} By multiplying equation (\ref{1}) by $u^{\prime }$ and integrate over $% \Omega $, using integrate by parts and summing up the product results, \begin{equation*} E\left( t\right) -E\left( 0\right) =-\int_{0}^{t}\left\vert \nabla u^{\prime }\left( s\right) \right\vert ^{2}ds-\alpha \int_{0}^{t}\int_{\Omega }u^{\prime }\left( s\right) g\left( u^{\prime }\left( s\right) \right) dxds% \text{ for }t\geq 0. \end{equation*} \end{proof} \begin{lemma} Suppose that (\ref{H1})-(\ref{H4}) hold, let $u_{0}\in W$ and $u_{1}\in H_{0}^{2}(\Omega )$ such that% \begin{equation} \gamma =\beta C_{\ast }^{\theta +1}\left( \frac{2p}{p-2}E\left( 0\right) \right) ^{\frac{\theta -1}{2}}\left( l_{1}+l_{1}\left\vert \left\vert k_{2}\left( x\right) \right\vert \right\vert _{\infty }+\left\vert \left\vert k_{1}\left( x\right) \right\vert \right\vert _{\infty }\right) <1. \label{9g} \end{equation}% Then $u\in W$ for each $t\geq 0.$ \end{lemma} Where $C_{\ast }$ is the Sobolev--Poincar\'{e} embedding such that for all $% 20$, by the continuity, there exists $00\ $for all $t\in \left[ 0,T_{m}\right] .$ By repeating this procedure, and using the fact that \begin{equation*} \underset{t\rightarrow T_{m}}{\lim }\beta C_{\ast }^{\theta +1}\left( \frac{% 2p}{p-2}E\left( t\right) \right) ^{\frac{\theta -1}{2}}\left( l_{1}+l_{1}\left\vert \left\vert k_{2}\left( x\right) \right\vert \right\vert _{\infty }+\left\vert \left\vert k_{1}\left( x\right) \right\vert \right\vert _{\infty }\right) \leq D<1, \end{equation*}% $T_{m}$ is extended to $T.$ \end{proof} \begin{lemma} Let the assumptions (\ref{9g}) holds. Then there exists $\eta =1-\gamma $ such that% \begin{equation} \beta \int_{\Omega }f\left( u\right) udx+\beta \int_{\Omega }k_{1}\left( x\right) \left\vert u\right\vert dx\leq \left( 1-\eta \right) \left\vert \Delta u\right\vert ^{2}, \label{13g} \end{equation}% and therefore% \begin{equation} \left\vert \Delta u\right\vert ^{2}\leq \frac{1}{\eta }I\left( t\right) . \label{14g} \end{equation} \end{lemma} \begin{proof} From (\ref{12g}) we have \begin{equation*} \beta \int_{\Omega }f\left( u\right) udx+\beta \int_{\Omega }k_{1}\left( x\right) \left\vert u\right\vert dx\leq \gamma \left\vert \Delta u\right\vert ^{2}. \end{equation*}% We get (\ref{13g}) by\ taking $\eta =$ $1-\gamma >0,$ and by using (\ref{13g}% ), from (\ref{2g}) we get the result (\ref{14g}). \end{proof} \begin{theorem} Suppose that (\ref{H1})-(\ref{H4}) hold. Let $u_{0}\in W$ satisfying (\ref% {9g}). Then the solution of problem (\ref{1}) is global. \end{theorem} \begin{proof} It sufficient to show that $\left\Vert u_{t}\right\Vert _{2}^{2}+\left\vert \Delta u\right\vert ^{2}$ is bounded independently to $t.$ To see this we use (\ref{2g}), (\ref{4g}), and (\ref{H3}) to obtain \begin{gather*} E(0)\geq E(t)=\frac{1}{2}\left\vert \Delta u\right\vert ^{2}-\beta \int_{\Omega }F\left( x,u\right) dx+\frac{\text{ }1}{2}\left\Vert u^{\prime }(t)\right\Vert _{2}^{2}+\frac{\text{ }1}{p}\left\Vert u(t)\right\Vert _{p}^{p} \\ \geq \frac{1}{2}\left\vert \Delta u\right\vert ^{2}-\frac{\beta }{p}% \int\limits_{\Omega }f\left( u\right) udx-\frac{\beta }{p}% \int\limits_{\Omega }k_{1}\left( x\right) \left\vert u\right\vert dx \\ +\frac{\text{ }1}{2}\left\Vert u^{\prime }(t)\right\Vert _{2}^{2}+\frac{% \text{ }1}{p}\left\Vert u(t)\right\Vert _{p}^{p}=\frac{1}{2}\left\vert \Delta u\right\vert ^{2}+\frac{1}{p}\left( I\left( t\right) -\left\vert \Delta u\right\vert ^{2}\right) + \\ +\frac{\text{ }1}{2}\left\Vert u^{\prime }(t)\right\Vert _{2}^{2}+\frac{% \text{ }1}{p}\left\Vert u(t)\right\Vert _{p}^{p} \\ =\frac{p-2}{2p}\left\vert \Delta u\right\vert ^{2}+\frac{1}{p}I\left( t\right) +\frac{\text{ }1}{2}\left\Vert u^{\prime }(t)\right\Vert _{2}^{2}+% \frac{\text{ }1}{p}\left\Vert u(t)\right\Vert _{p}^{p} \\ \geq \frac{\text{ }1}{2}\left\Vert u^{\prime }(t)\right\Vert _{2}^{2}+\frac{% p-2}{2p}\left\vert \Delta u\left( t\right) \right\vert ^{2}, \end{gather*}% since $I\left( t\right) \geq 0,$ and $p>2$. Therefore% \begin{equation*} \left\Vert u^{\prime }(t)\right\Vert _{2}^{2}+\left\vert \Delta u\right\vert ^{2}\leq \max \left( 2,\frac{2p}{p-2}\right) E(0). \end{equation*}% These estimates imply that the solution $u(t)$ exist globally in $\left[ 0,+\infty \right[ $. \end{proof} \section{Blow-up of Solution} In this section, after some estimates, we show that the solution of problem (% \ref{1}) blows up in finite time under the assumption $E(0)<0$, where \begin{equation} E(t)=E(u(t),u^{\prime }(t))=\frac{1}{2}\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\frac{1}{2}\left\vert \Delta u\left( t\right) \right\vert ^{2}+\frac{1}{p}\left\Vert u(t)\right\Vert _{p}^{p}-\beta \int_{\Omega }F\left( x,u\left( t\right) \right) dx. \label{1s} \end{equation} \begin{remark} We set \begin{equation} H\left( t\right) =-E\left( t\right) , \label{3s} \end{equation}% we multiply Eq.(\ref{1}) by $-u^{\prime }$ and integrate over $\Omega ,$ using (\ref{H2}) to get% \begin{equation} H^{\prime }\left( t\right) =\left\vert \nabla u^{\prime }\left( t\right) \right\vert ^{2}+\alpha \int_{\Omega }u^{\prime }\left( t\right) g\left( u^{\prime }\left( t\right) \right) dx\geq \alpha d_{0}\left\Vert u^{\prime }\left( t\right) \right\Vert _{\sigma }^{\sigma }\text{ a.e. }t\in \left[ 0,T% \right] , \label{5sb} \end{equation}% $H\left( t\right) $ is absolutely continuous, hence% \begin{equation} 01$, depending only on $% \Omega $, such that% \begin{equation} \left\Vert u(t)\right\Vert _{p}^{s}\leq C\left( \left\Vert u(t)\right\Vert _{p}^{p}+\left\vert \Delta u\left( t\right) \right\vert ^{2}\right) \text{, with }2\leq s\leq p, \label{2s} \end{equation}% for any $u\in H_{0}^{2}\left( \Omega \right) $. If $u$ is the solution constructed in Theorem \ref{theorem1}, then% \begin{equation} \left\Vert u(t)\right\Vert _{p}^{s}\leq C\left( H\left( t\right) +\left\Vert u(t)\right\Vert _{p}^{p}+\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\beta \int_{\Omega }F\left( x,u\left( t\right) \right) dx\right) , \label{4s} \end{equation}% with $2\leq s\leq p$ on $\left[ 0,T\right) .$ \end{lemma} \begin{theorem} Let the conditions of the Theorem \ref{theorem1} be satisfied. Assume further that% \begin{equation} E\left( 0\right) <0. \label{5s} \end{equation}% Then the solution (\ref{6b}) blows up in a finite time $T$. \end{theorem} \begin{proof} We pose \begin{equation*} \left\{ \begin{array}{c} L\left( t\right) =\left\vert u\left( t\right) \right\vert ^{2}=\int\limits_{\Omega }\left\vert u\left( x,t\right) \right\vert ^{2}dx, \\ L^{\prime }\left( t\right) =2\left( u\left( t\right) ,u^{\prime }\left( t\right) \right) , \\ L^{\prime \prime }\left( t\right) =2\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+2\left( u\left( t\right) ,u^{\prime \prime }\left( t\right) \right) ,% \end{array}% \right. \end{equation*}% we define the function \begin{gather} G\left( t\right) =H^{1-a}\left( t\right) +\varepsilon L^{\prime }\left( t\right) -3\varepsilon pe^{T-t}\beta \int_{\Omega }F\left( x,u\left( t\right) \right) dx \notag \\ +\gamma _{1}\varepsilon t\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\gamma _{2}\varepsilon t\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma },\text{ }t\geq 0, \label{7s} \end{gather}% where $\gamma _{1},$ $\gamma _{2},$ $\varepsilon >0$ are positives constants to be specified later, and% \begin{equation} 00, \notag \end{gather}% because $\frac{\sigma }{\sigma -1}\leq \sigma ,$ then by (\ref{5sb}) we have \begin{eqnarray} &&d_{1}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\frac{\sigma }{\sigma -1}}^{\frac{\sigma }{\sigma -1}% }+d_{2}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }\leq C\left( \Omega \right) ^{\frac{\sigma -2% }{\sigma }}d_{1}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\sigma }^{\frac{\sigma }{\sigma -1}}+\frac{d_{2}}{\alpha d_{0}}% H^{\prime }\left( t\right) \notag \\ &\leq &C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{\sigma -2}{\sigma }% }\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }+\frac{d_{2}}{\alpha d_{0}}H^{\prime }\left( t\right) \leq \frac{1}{\alpha d_{0}}\left( C^{\ast }d_{1}C\left( \Omega \right) ^{% \frac{\sigma -2}{\sigma }}+d_{2}\right) H^{\prime }\left( t\right) . \label{10sb} \end{eqnarray}% By the boundary conditions we derive the following estimates \begin{equation} \int_{\Omega }u\left( t\right) \Delta u^{\prime }\left( t\right) dx=\int_{\Omega }\Delta u\left( t\right) u^{\prime }\left( t\right) dx\leq \frac{1}{4}\left\vert \Delta u\left( t\right) \right\vert ^{2}+\left\vert u^{\prime }\left( t\right) \right\vert ^{2}. \label{11sb} \end{equation}% Using hypotheses (\ref{H4}), Holder's, Young's inequalities, conditions (\ref% {8s}), and (\ref{5sb}) we have \begin{gather*} \int\limits_{\Omega }\left\vert f\left( u\left( t\right) \right) \right\vert \left\vert u^{\prime }\left( t\right) \right\vert dx\leq l_{1}\int\limits_{\Omega }\left( \left\vert u\right\vert ^{\theta }\left\vert u^{\prime }\left( t\right) \right\vert +\left\vert k_{2}\left( x\right) \right\vert \left\vert u^{\prime }\left( t\right) \right\vert \right) dx \\ \leq l_{1}\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{2\theta }^{\theta }\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{2} \\ +l_{1}\frac{\sigma -1}{\sigma }\delta ^{\frac{\sigma }{1-\sigma }}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\frac{\sigma }{\sigma -1}}^{\frac{\sigma }{\sigma -1}}+l_{1}\frac{\delta ^{\sigma }}{% \sigma }\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \\ \leq \frac{l_{1}}{\sigma }C\left( \delta ,\sigma \right) \delta ^{\sigma }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{2\theta }^{2\theta }+\frac{1}{\sigma }l_{1}\delta ^{\frac{\sigma }{1-\sigma }% }\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{2}^{2} \\ +l_{1}\frac{\sigma -1}{\sigma }\delta ^{\frac{\sigma }{1-\sigma }}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\frac{\sigma }{\sigma -1}}^{\frac{\sigma }{\sigma -1}}+l_{1}\frac{\delta ^{\sigma }}{% \sigma }\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \\ \leq \frac{l_{1}}{\sigma }C^{\ast }C\left( \delta ,\sigma \right) C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }}\left\vert \left\vert u\right\vert \right\vert _{\sigma }^{\sigma } \\ +\frac{1}{\sigma }l_{1}C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{% 2\sigma }}\delta ^{\frac{\sigma }{1-\sigma }}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma } \\ +l_{1}\frac{\sigma -1}{\sigma }\delta ^{\frac{\sigma }{1-\sigma }}C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{2\sigma }}\left\vert \left\vert u^{\prime }\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }+l_{1}% \frac{\delta ^{\sigma }}{\sigma }\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \\ \leq \frac{l_{1}}{\alpha d_{0}}C^{\ast }C\left( \Omega \right) ^{\frac{% \sigma -2}{2\sigma }}\delta ^{\frac{\sigma }{1-\sigma }}H^{\prime }\left( t\right) \\ +\frac{l_{1}}{\sigma }C\left( \delta ,\sigma \right) \delta ^{\sigma }C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }% }\left\vert \left\vert u\right\vert \right\vert _{\sigma }^{\sigma }+l_{1}% \frac{\delta ^{\sigma }}{\sigma }\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }. \end{gather*}% By the hypotheses (\ref{H3}), and the estimate (\ref{6s}) we have% \begin{eqnarray} 2\beta \int\limits_{\Omega }u\left( t\right) f\left( u\left( t\right) \right) dx &\geq &2\beta p\int\limits_{\Omega }F\left( x\right) dx-2\beta \int\limits_{\Omega }k_{1}\left( x\right) \left\vert u\left( x\right) \right\vert dx \notag \\ &\geq &2pH\left( t\right) -2\beta \int\limits_{\Omega }k_{1}\left( x\right) \left\vert u\left( x\right) \right\vert dx, \label{12s} \end{eqnarray}% and by Holder's, Young's inequalities, \begin{equation} \int\limits_{\Omega }k_{1}\left( x\right) \left\vert u\left( x\right) \right\vert dx\leq C\left( \sigma ,\alpha \right) \left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+2\alpha \frac{\delta ^{\sigma }}{\sigma }% \left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }. \label{12bs} \end{equation}% By substituting in (\ref{9s}), and using (\ref{10s})-(\ref{12bs}), yields, \begin{gather} \frac{d}{dt}G\left( t\right) \notag \\ \geq \left( \begin{array}{c} \left( 1-a\right) H^{-a}\left( t\right) \\ -\frac{1}{\alpha d_{0}}\left( 3p\varepsilon e^{T-t}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{2\sigma }}+2\alpha \varepsilon \frac{% \sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{% \sigma -2}{\sigma }}+d_{2}\right) \right) \delta ^{\frac{\sigma }{1-\sigma }}% \end{array}% \right) H^{\prime }\left( t\right) \notag \\ +2p\varepsilon H\left( t\right) -2\varepsilon \left\Vert u\left( t\right) \right\Vert _{p}^{p}-\frac{5}{2}\varepsilon \left\vert \Delta u\left( t\right) \right\vert ^{2}+\left( \gamma _{1}-2\beta C\left( \sigma ,\alpha \right) \right) \varepsilon \left\Vert k_{1}\left( x\right) \right\Vert _{\infty } \notag \\ +\left( \gamma _{2}-3p\varepsilon e^{T-t}\beta l_{1}\frac{\delta ^{\sigma }}{% \sigma }\right) \varepsilon \left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }+3p\beta \varepsilon \int\limits_{\Omega }F\left( x,u\left( s\right) \right) dx \notag \\ -\varepsilon \left( 3\theta pe^{T-t}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }}+2\beta \alpha \left( d_{1}+d_{2}\right) \right) \frac{\delta ^{\sigma }}{\sigma }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma },\text{ }\forall \delta ,\text{ }\varepsilon >0. \label{13s} \end{gather}% At this point, for a large positive constant $\lambda $ to be chosen later, picking $\delta $ such that $\delta ^{\frac{\sigma }{1-\sigma }}=\lambda H^{-a}\left( t\right) >0$ in (\ref{13s}) we arrive for all $t>0$ at% \begin{gather} \frac{d}{dt}G\left( t\right) \notag \\ \geq \left( \begin{array}{c} \left( 1-a\right) \\ -\frac{\lambda }{\alpha d_{0}}\left( 3p\varepsilon e^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{2\sigma }}+2\alpha \varepsilon \frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{% \sigma -2}{\sigma }}+d_{2}\right) \right)% \end{array}% \right) H^{-a}\left( t\right) H^{\prime }\left( t\right) \notag \\ +3\beta p\varepsilon \int_{\Omega }F\left( x,u\right) dx-2\varepsilon \left\Vert u\left( t\right) \right\Vert _{p}^{p}-\frac{5}{2}\varepsilon \left\vert \Delta u\left( t\right) \right\vert ^{2}+2p\varepsilon H\left( t\right) \notag \\ +\left( \gamma _{1}-2\beta C\left( \sigma ,\alpha \right) \right) \varepsilon \left\Vert k_{1}\left( x\right) \right\Vert _{\infty } \label{14s} \\ +\left( \gamma _{2}-3p\varepsilon e^{T}\beta l_{1}\frac{\delta ^{\sigma }}{% \sigma }\right) \varepsilon \left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \notag \\ -\varepsilon \left( 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }% }+2\beta \alpha \left( d_{1}+d_{2}\right) \right) \frac{\lambda ^{1-\sigma }% }{\sigma }H^{a\left( \sigma -1\right) }\left( t\right) \left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma },\text{ }% \forall \delta ,\text{ }\varepsilon >0. \notag \end{gather}% By exploiting (\ref{6s}), we have% \begin{equation} H^{a\left( \sigma -1\right) }\left( t\right) \left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }\leq \beta ^{a\left( \sigma -1\right) }\left( \int\limits_{\Omega }F\left( x,u\right) dx\right) ^{a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }, \label{15s} \end{equation}% from (\ref{H3}) we have% \begin{gather} \int_{\Omega }F\left( x,u\right) dx\leq \frac{l_{1}}{p}\left( \int\limits_{\Omega }\left\vert u\left( t\right) \right\vert ^{\theta +1}dx+\left( \left\vert k_{2}\left( x\right) \right\vert +\left\vert k_{1}\left( x\right) \right\vert \right) \left\vert u\right\vert \right) \notag \\ \leq \frac{l_{1}}{p}\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\theta +1}+C\frac{l_{1}}{p}\left( \left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }\right) \left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\theta +1}, \notag \\ \leq C\frac{l_{1}}{p}\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\theta +1} \label{16s} \end{gather}% by condition (\ref{8s}), and the estimates (\ref{4s}) we confirm that% \begin{gather} \beta ^{a\left( \sigma -1\right) }\left\vert \int\limits_{\Omega }F\left( x,u\right) dx\right\vert ^{a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma }\leq \notag \\ \leq C\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\left( \left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\theta +1}\right) ^{a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma } \notag \\ =C\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\left( \theta +1\right) a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\sigma }^{\sigma } \notag \\ \leq C\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\left( \theta +1\right) a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\sigma } \notag \\ =C\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\left\vert \left\vert u\left( t\right) \right\vert \right\vert _{\theta +1}^{\left( \theta +1\right) a\left( \sigma -1\right) +\sigma } \notag \\ \leq \frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }C\left( H\left( t\right) +\left\Vert u(t)\right\Vert _{p}^{p}+\left\vert u^{\prime }(t)\right\vert ^{2}+\beta \int\limits_{\Omega }F\left( x,u\right) dx\right) \notag \\ \leq C\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\left( \begin{array}{c} H\left( t\right) +\left\Vert u(t)\right\Vert _{p}^{p}+\left\vert u^{\prime }(t)\right\vert ^{2}+\beta \int\limits_{\Omega }F\left( x,u\right) dx \\ +\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }% \end{array}% \right) \label{17s} \end{gather}% substituting (\ref{17s}) in (\ref{14s}) we obtain \begin{gather} \frac{d}{dt}G\left( t\right) \notag \\ \geq \left( \left( 1-a\right) -\frac{\lambda }{\alpha d_{0}}\left( \begin{array}{c} 3p\varepsilon e^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{% 2\sigma }} \\ +2\alpha \varepsilon \frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{\sigma -2}{\sigma }}+d_{2}\right)% \end{array}% \right) \right) \notag \\ \times H^{-a}\left( t\right) H^{\prime }\left( t\right) \notag \\ +3p\beta \varepsilon \int\limits_{\Omega }F\left( x,u\right) dx-\frac{5}{2}% \varepsilon \left\vert \Delta u\left( t\right) \right\vert ^{2}-2\varepsilon \left\Vert u\left( t\right) \right\Vert _{p}^{p} \notag \\ +\varepsilon \left( \gamma _{1}-2\beta C\left( \sigma ,\alpha \right) \right) \left\Vert k_{1}\left( x\right) \right\Vert _{\infty } \notag \\ +\varepsilon \left( \gamma _{2}-3p\varepsilon e^{T}\beta l_{1}\frac{\delta ^{\sigma }}{\sigma }\right) \left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \label{20s} \\ +\varepsilon \left( \begin{array}{c} 2pH\left( t\right) -\left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) } \\ \times C\left( \begin{array}{c} H\left( t\right) +\left\Vert u(t)\right\Vert _{p}^{p}+\left\vert u^{\prime }(t)\right\vert ^{2}+\beta \int\limits_{\Omega }F\left( x,u\right) dx \\ +\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }% \end{array}% \right)% \end{array}% \right) \notag \end{gather}% or \begin{gather} \frac{d}{dt}G\left( t\right) \notag \\ \geq \left( \begin{array}{c} \left( 1-a\right) \\ -\frac{\lambda }{\alpha d_{0}}\left( \begin{array}{c} 3p\varepsilon e^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{% 2\sigma }} \\ +2\alpha \varepsilon \frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{\sigma -2}{\sigma }}+d_{2}\right)% \end{array}% \right)% \end{array}% \right) H^{-a}\left( t\right) H^{\prime }\left( t\right) \notag \\ +3p\beta \varepsilon \int_{\Omega }F\left( x,u\right) dx-\frac{5}{2}% \varepsilon \left\vert \Delta u\left( t\right) \right\vert ^{2}-2\varepsilon \left\Vert u\left( t\right) \right\Vert _{p}^{p} \notag \\ +\varepsilon \left( \gamma _{1}-2\beta C\left( \sigma ,\alpha \right) \right) \left\Vert k_{1}\left( x\right) \right\Vert _{\infty } \label{21s} \\ +\varepsilon \left( \gamma _{2}-3p\varepsilon e^{T}\beta l_{1}\frac{\delta ^{\sigma }}{\sigma }\right) \left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \notag \\ +\varepsilon \left( \begin{array}{c} \left( 5p-1\right) H\left( t\right) \\ -\left( \begin{array}{c} 3\theta pe^{T-t}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) } \\ \times C\left( \begin{array}{c} H\left( t\right) +\left\Vert u(t)\right\Vert _{p}^{p}+\left\vert u^{\prime }(t)\right\vert ^{2}+\beta \int_{\Omega }F\left( x,u\right) dx \\ +\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }% \end{array}% \right)% \end{array}% \right) \notag \\ -\varepsilon \left( 3p-1\right) H\left( t\right) . \notag \end{gather}% By using the definition (\ref{3s}), the estimate (\ref{21s}) gives% \begin{gather*} \frac{d}{dt}G\left( t\right) \\ \geq \left( \begin{array}{c} \left( 1-a\right) \\ -\frac{\lambda }{\alpha d_{0}}\left( \begin{array}{c} 3p\varepsilon e^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{% 2\sigma }} \\ +2\alpha \varepsilon \frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{\sigma -2}{\sigma }}+d_{2}\right)% \end{array}% \right)% \end{array}% \right) \\ \times H^{-a}\left( t\right) H^{\prime }\left( t\right) \end{gather*}% \begin{gather*} +\varepsilon \left[ \begin{array}{c} \left( \frac{3p-1}{2}\right) \\ -\left( C\left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right)% \end{array}% \right] \\ \times \left\vert u^{\prime }\left( t\right) \right\vert ^{2} \end{gather*}% \begin{gather*} +\left( \frac{3p-1}{2}-\frac{5}{2}\right) \varepsilon \left\vert \Delta u\left( t\right) \right\vert ^{2} \\ +\varepsilon \left[ \begin{array}{c} \left( \gamma _{1}-2\beta C\left( \sigma ,\alpha \right) \right) \\ -C\left( \left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right)% \end{array}% \right] \left\Vert k_{1}\left( x\right) \right\Vert _{\infty } \end{gather*}% \begin{equation*} +\varepsilon \left[ \begin{array}{c} \left( \gamma _{2}-3p\varepsilon e^{T}\beta l_{1}\frac{\delta ^{\sigma }}{% \sigma }\right) \\ -C\left( \left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right)% \end{array}% \right] \left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \end{equation*}% \begin{gather*} +\varepsilon \left[ \begin{array}{c} \left( \frac{3p-1}{p}-2\right) \\ -C\left( \left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right)% \end{array}% \right] \\ \times \left\Vert u\left( t\right) \right\Vert _{p}^{p} \end{gather*}% \begin{equation*} +\varepsilon \left[ \begin{array}{c} 3p-\left( 3p-1\right) \\ -C\left( \left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right)% \end{array}% \right] \beta \int\limits_{\Omega }F\left( x,u\right) dx \end{equation*}% \begin{equation*} +\varepsilon \left[ \begin{array}{c} \left( 5p-1\right) \\ -C\left( \left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{\lambda ^{1-\sigma }}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right)% \end{array}% \right] H\left( t\right) . \end{equation*}% pose% \begin{equation*} C_{1}=C\left( \left( \begin{array}{c} 3\theta pe^{T}\beta l_{1}C\left( \delta ,\sigma \right) C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2\theta }{2\theta \sigma }} \\ +2\beta \alpha \left( d_{1}+d_{2}\right)% \end{array}% \right) \frac{1}{\sigma }\frac{l_{1}}{p}\beta ^{a\left( \sigma -1\right) }\right) , \end{equation*}% we arrive at% \begin{gather} \frac{d}{dt}G\left( t\right) \notag \\ \geq \left( \begin{array}{c} \left( 1-a\right) \\ -\frac{\lambda }{\alpha d_{0}}\varepsilon \left( \begin{array}{c} 3pe^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{2\sigma }} \\ +2\frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{% \frac{\sigma -2}{\sigma }}+d_{2}\right)% \end{array}% \right)% \end{array}% \right) H^{-a}\left( t\right) H^{\prime }\left( t\right) \notag \\ +\varepsilon \left[ \frac{3p-1}{2}-C_{1}\lambda ^{1-\sigma }\right] \left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\left( \frac{3p-1}{2}-% \frac{5}{2}\right) \varepsilon \left\vert \Delta u\left( t\right) \right\vert ^{2} \notag \\ +\varepsilon \left( \left( \gamma _{1}-2\beta C\left( \sigma ,\alpha \right) \right) -C_{1}\lambda ^{1-\sigma }\right) \left\Vert k_{1}\left( x\right) \right\Vert _{\infty } \notag \\ +\varepsilon \left( \left( \gamma _{2}-3p\varepsilon e^{T}\beta l_{1}\frac{% \delta ^{\sigma }}{\sigma }\right) -C_{1}\lambda ^{1-\sigma }\right) \left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma } \notag \\ +\varepsilon \left[ \frac{p-1}{p}-C_{1}\lambda ^{1-\sigma }\right] \left\Vert u\left( t\right) \right\Vert _{p}^{p}+\varepsilon \left[ 1-C_{1}\lambda ^{1-\sigma }\right] \beta \int\limits_{\Omega }F\left( x,u\right) dx \label{22s} \\ +\varepsilon \left( \left( 5p-1\right) -C_{1}\lambda ^{1-\sigma }\right) H\left( t\right) . \notag \end{gather}% chosen $\gamma _{1}=1+2\beta C\left( \sigma ,\alpha \right) ,\gamma _{2}=1+3p\varepsilon e^{T}\beta l_{1}\frac{\delta ^{\sigma }}{\sigma }$ and $% \lambda $ satisfying the following inequality% \begin{equation*} \lambda \geq \lambda _{0}=\min \left( \sqrt[\sigma -1]{\frac{2C_{1}}{3p-1}},% \sqrt[\sigma -1]{\frac{pC_{1}}{p-1}},\sqrt[\sigma -1]{C_{1}},\sqrt[\sigma -1]% {\frac{C_{1}}{5p-1}}\right) \end{equation*}% so that the coefficients of $H\left( t\right) ,$ $\left\vert u^{\prime }\left( t\right) \right\vert ^{2},$ $\left\vert \Delta u\left( t\right) \right\vert ^{2},$ $\left\Vert u\left( t\right) \right\Vert _{p}^{p}$ , $% \left\Vert k_{1}\left( x\right) \right\Vert _{\infty }$, $\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }$ and $\int\limits_{\Omega }F\left( x,u\right) dx$ in (\ref{22s}) are strictly positive, hence we get \begin{gather} \frac{d}{dt}G\left( t\right) \notag \\ \geq \left( \begin{array}{c} \left( 1-a\right) \\ -\frac{\lambda }{\alpha d_{0}}\varepsilon \left( \begin{array}{c} 3pe^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{2\sigma }} \\ +2\alpha \frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{\sigma -2}{\sigma }}+d_{2}\right)% \end{array}% \right)% \end{array}% \right) H^{-a}\left( t\right) H^{\prime }\left( t\right) \notag \\ +\omega \varepsilon \left( \begin{array}{c} H\left( t\right) +\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\left\Vert u\left( t\right) \right\Vert _{p}^{p}+\int_{\Omega }F\left( x,u\right) dx \\ +\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }% \end{array}% \right) , \label{23s} \end{gather}% where $\omega $ is the minimum of these coefficients. We pick $\varepsilon $ small enough, so that \begin{equation*} 0<\varepsilon \leq \varepsilon _{0}=\min \left( \begin{array}{c} \frac{1-a}{\frac{\lambda }{\alpha d_{0}}\left( \begin{array}{c} 3pe^{T}\beta C^{\ast }C\left( \Omega \right) ^{\frac{\sigma -2}{2\sigma }} \\ +2\alpha \frac{\sigma -1}{\sigma }\left( C^{\ast }d_{1}C\left( \Omega \right) ^{\frac{\sigma -2}{\sigma }}+d_{2}\right)% \end{array}% \right) }; \\ \frac{H^{1-a}\left( 0\right) }{-L^{\prime }\left( 0\right) +3pe^{T}\beta \int_{\Omega }F\left( x,u_{0}\right) dx}% \end{array}% \right) \end{equation*}% therefore (\ref{23s}) take the form \begin{equation} \frac{d}{dt}G\left( t\right) \geq \omega \varepsilon \left( \begin{array}{c} H\left( t\right) +\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\left\Vert u\left( t\right) \right\Vert _{p}^{p} \\ +\int_{\Omega }F\left( x,u\right) dx+\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }% \end{array}% \right) , \label{24s} \end{equation}% hence% \begin{equation*} G\left( t\right) \geq G\left( 0\right) >0\text{ for all }t\geq 0. \end{equation*}% The second term in (\ref{7s}), by applying Young's inequality we can estimate as follows% \begin{equation*} \frac{1}{2}L^{\prime }\left( t\right) =\left( u\left( t\right) ,u^{\prime }\left( t\right) \right) \leq c\left\vert u^{\prime }\left( t\right) \right\vert \left\Vert u\left( t\right) \right\Vert _{p}\leq c\left( \left\vert u^{\prime }\left( t\right) \right\vert ^{2\left( 1-a\right) }+\left\Vert u\left( t\right) \right\Vert _{p}^{\frac{2\left( 1-a\right) }{% 1-2a}}\right) , \end{equation*}% so% \begin{equation*} \left\vert \left( u\left( t\right) ,u^{\prime }\left( t\right) \right) \right\vert ^{\frac{1}{1-a}}\leq C\left( \left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\left\Vert u\left( t\right) \right\Vert _{p}^{% \frac{2}{1-2a}}\right) \end{equation*}% using Lemma (\ref{Lemma1}) and the condition (\ref{8s}) we obtain \begin{gather} \left\vert \left( u\left( t\right) ,u^{\prime }\left( t\right) \right) \right\vert ^{\frac{1}{1-a}} \notag \\ \leq C\left( H\left( t\right) +\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\left\Vert u\left( t\right) \right\Vert _{p}^{p}+\int_{\Omega }F\left( x,u\right) dx\right) ,\text{ }\forall t\geq 0. \label{25s} \end{gather}% Consequently we have% \begin{gather} G\left( t\right) ^{\frac{1}{1-a}}=\left( H^{1-a}\left( t\right) +2\varepsilon \int_{\Omega }u\left( x,t\right) u^{\prime }\left( t\right) dx+\gamma _{1}\varepsilon t\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\gamma _{2}\varepsilon t\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }\right) ^{\frac{1}{1-a}} \notag \\ \leq c\left( H\left( t\right) +\left\vert 2\varepsilon \int_{\Omega }u\left( x,t\right) u^{\prime }\left( t\right) dx\right\vert ^{\frac{1}{1-a}% }+\left\vert \gamma _{1}\varepsilon t\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }\right\vert ^{^{\frac{1}{1-a}}}+\left\vert \gamma _{2}\varepsilon t\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }\right\vert ^{^{\frac{1}{1-a}}}\right) \notag \\ \leq C\left( H\left( t\right) +\left\vert u^{\prime }\left( t\right) \right\vert ^{2}+\left\Vert u\left( t\right) \right\Vert _{p}^{p}+\int_{\Omega }F\left( x,u\right) dx+\left\Vert k_{1}\left( x\right) \right\Vert _{\infty }+\left\Vert k_{2}\left( x\right) \right\Vert _{\infty }^{\sigma }\right) . \label{26s} \end{gather}% We then combine (\ref{24s}), (\ref{25s}), and (\ref{26s}), to arrive at \begin{equation} \frac{d}{dt}G\left( t\right) \geq \rho G\left( t\right) ^{\frac{1}{1-a}}, \label{27s} \end{equation}% where $\rho $ is a constant depending on $C,$ $\omega $, and $\varepsilon $ only, and not depend of $u.$ Integrate (\ref{27s}) over $\left( 0,t\right) $ to get% \begin{equation*} G\left( t\right) ^{\frac{a}{1-a}}\geq \frac{1}{G^{\frac{a-1}{a}}\left( 0\right) -t\frac{a}{\left( 1-a\right) }\rho }. \end{equation*}% Therefore $G$ $\left( t\right) $ blows up in a finite time $T^{\ast }$ where \begin{equation*} T^{\ast }\leq \frac{1-a}{a\rho G^{\frac{a}{1-a}}\left( 0\right) }. \end{equation*} \end{proof} \begin{thebibliography}{99} \bibitem{Adams} Adams, R.A, \emph{Sobolev Spaces}. 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