\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 297, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/297\hfil Exponential stability and blow-up] {Exponential stability and blow-up for abstract nonlinear system with source terms} \author[P. Wang, J. Hao \hfil EJDE-2017/297\hfilneg] {Peipei Wang, Jianghao Hao} \address{Peipei Wang \newline School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{1576037528@qq.com} \address{Jianghao Hao (corresponding author)\newline School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{hjhao@sxu.edu.cn} \dedicatory{Communicated by Paul Rabinowitz} \thanks{Submitted June 8, 2017. Published November 28, 2017.} \subjclass[2010]{35L05, 35L20, 35L70, 93D15} \keywords{Abstract nonlinear system; exponential decay; convex method; \hfill\break\indent blow-up} \begin{abstract} In this article we consider an abstract nonlinear system with nonlinear source terms. We prove the exponential stability by the energy method. Also under suitable conditions on the initial values, we show that the nonlinear source terms are able to guarantee the blow-up of the solutions by convex method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Let $A:D(A)\to L^2(\Omega)$ be a self-adjoint positive definite operator, $D(A)\subset L^2(\Omega)$ is a dense and compact embedding where $\Omega$ is a open bounded subset of $\mathbb{R}^n~(n\geq1)$. We consider the system \begin{equation} \begin{gathered} u_{tt}+A^2u+M(\|A^{\alpha/2}u\|_2^2 +\|A^{\alpha/2}v\|_2^2)A^{\alpha}u+N(\|A^{\beta/2}u\|_2^2)A^{\beta}u_t=f(u,v),\\ \text{in } \Omega\times(0,\infty),\\ v_{tt}+A^2v+M(\|A^{\alpha/2}u\|_2^2+\|A^{\alpha/2}v\|_2^2)A^{\alpha}v +N(\|A^{\beta/2}v\|_2^2)A^{\beta}v_t=g(u,v), \\ \text{in } \Omega\times(0,\infty),\\ \end{gathered} \label{e1.1} \end{equation} with the initial value conditions \begin{equation} \begin{gathered} u(x,0)=u_0,v(x,0)=v_0,\quad\text{in } \Omega,\\ u_t(x,0)=u_1,v_t(x,0)=v_1,\quad\text{in } \Omega,\\ \end{gathered} \label{e1.2} \end{equation} where $0<\beta\leq\alpha\leq1$, $M$ and $N$ are continuous functions. The functions $f$ and $g$ model the interior dissipations in the equations. Many authors have studied the nonlinear wave equation \begin{equation} u_{tt}-M\left(\|\nabla u\|_2^2\right)\Delta u=0. \label{e1.3} \end{equation} This model was proposed by Kirchhoff \cite{K} in one dimensional in which $M(s)$ is a linear function, and describe the transversal vibration of a string. Then many authors devote themselves to local existence and global existence result of system \eqref{e1.3}, see \cite{DS,DS-1,EMM,M}. Mizumachi \cite{M-1} added linear damping on \eqref{e1.3} and obtained the decay estimates for the solutions. Ikehata \cite{I} considered the local solvability for abstract equation $$ u_{tt}-M\big(\|A^{1/2}u\|^2\big)Au+\delta u_t=f(u), $$ where $A$ is a positive definite and self-adjoint operator in Hilbert space $(X, \|\cdot\|)$, $f:D(A^{1/2})\to X$ is a nonlinear operator, and $M(s)$ is a $C^1$ function satisfying $$ M(s)\geq m_0>0. $$ He obtained the existence of strong solution without compactness hypothesis. Rivera \cite{R} studied the equation with damping $$\begin{array}{lll} u_{tt}+M\left(\|A^{1/2}u\|^2\right)Au+Au_t=0,\\ \end{array}$$ and proved that if the initial value $(u_0, u_1)\in D(A)\times X$, then the corresponding solution of the system satisfies $$ u\in C^2\big([0,T];D(A^k)\big), \quad \forall k\in N. $$ Lazo \cite{LP} studied the nonlinear wave equation $$ u_{tt}+M\big(\|A^{1/2}u\|^2\big)Au +N\left(\|A^{\alpha}u\|^2\right) A^{\alpha}u_t=f. $$ He proved the existence of global solutions in a Hilbert space by using Galerkin's method. Wu \cite{W} considered the following nonlinear viscoelastic wave equations of Kirchhoff type with the nonlinear damping and source terms, \begin{gather*} u_{tt}-M\left(\|\nabla u\|_2^2+\|\nabla v\|_2^2\right)\Delta u +\int^t_0g_1(t-s)\Delta u(s)ds+|u_t|^{p-1}u_t=f_1(u,v),\\ v_{tt}-M\left(\|\nabla u\|_2^2+\|\nabla v\|_2^2\right)\Delta v +\int^t_0g_2(t-s)\Delta v(s)ds+|v_t|^{q-1}v_t=f_2(u,v). \end{gather*} He proved that, with the initial date in the stable set and for a wider class of relaxation functions, the decay rate of the system depends on the exponents of the damping terms by using Nakao's method. Conversely, for certain initial date in the unstable set, he obtained the blow-up result when the initial energy is nonnegative. Mu and Ma \cite{MM} considered the following nonlinear viscoelastic wave equations of Kirchhoff type with Balakrishnan-Taylor damping, \begin{gather*} u_{tt}-\Big(a+b\|\nabla u\|_2^2+\sigma\int_{\Omega}\nabla u\cdot\nabla u_t\,dx\Big) \Delta u+\int^t_0g_1(t-s)\Delta u(s)ds=f_1(u,v),\\ v_{tt}-\Big(a+b\|\nabla v\|_2^2+\sigma\int_{\Omega}\nabla v\cdot\nabla v_t\,dx\Big) \Delta v+\int^t_0g_2(t-s)\Delta v(s)ds=f_2(u,v). \end{gather*} By the modified perturbed energy technique, the authors showed that the decay rate of the system is similar to that of relaxation functions. They also proved that nonlinear source of polynomial type is able to force solutions to blow up in finite time even if stronger damping exists. Zhang et al \cite{ZDZ} obtained the existence of global weak solutions for the coupled system \begin{gather*} u_{tt}+M(\|A^{1/2}u\|^2+\|A^{1/2}v\|^2)Au +N(\|A^{\alpha}u\|^2)A^{\alpha}u_t=f(x,t),\\ v_{tt}+M(\|A^{1/2}u\|^2+\|A^{1/2}v\|^2)Av +N(\|A^{\alpha}v\|^2)A^{\alpha}v_t=g(x,t). \end{gather*} Hao et al \cite{ZJW} proved the well posedness of the solution for system \begin{gather*} u_{tt}+A^2u+M(\|A^{\alpha/2}u\|^2+\|A^{\alpha/2}v\|^2)A^{\alpha}u +N(\|A^{\beta/2}u\|^2)A^{\beta}u_t=f(x, t),\\ v_{tt}+A^2v+M(\|A^{\alpha/2}u\|^2+\|A^{\alpha/2}v\|^2)A^{\alpha}v +N(\|A^{\beta/2}v\|^2)A^{\beta}v_t=g(x, t), \end{gather*} by Galerkin's method. However, they did not obtain the blow up or decay property. In this paper, under suitable conditions, we prove that the system is exponential stable when the initial value lies in the stable set and blow up when the initial energy is negative or non-negative but small. Our plan in this paper is as follows. In section 2, we present some materials and assumptions needed later. In section 3, we prove the decay result by the energy method. In section 4, by using the convex method, we prove the blow-up phenomena of solutions. \section{Preliminaries} In this section, we present some materials and assumptions needed in the rest of this paper. We denote $\|\cdot\|_q=\|\cdot\|_{L^q(\Omega)}$, $1\leq q<\infty$, and denote $(\cdot,\cdot)$ the usual inner product of $L^2(\Omega)$. We denote $H=L^2(\Omega)$ and $c$ to be a generic positive constant which might change from line to line. Next we give some assumptions for system \eqref{e1.1}--\eqref{e1.2}. \begin{itemize} \item[(A1)] There exist constants $c_0>0,\gamma>2$, $p>1$ and a positive $C^1$ function $F:R^2\to R$ such that \begin{equation} \begin{gathered} \frac{\partial F(u,v)}{\partial u}=f(u,v),\quad \frac{\partial F(u,v)}{\partial v}=g(u,v),\quad uf(u,v)+vg(u,v)=\gamma F, \\ \int_\Omega F(u,v)dx\leq c_0\left(\|u\|_{p+1}^{p+1}+\|v\|_{p+1}^{p+1}\right), \end{gathered} \label{e2.1} \end{equation} where $p$ satisfies the inequality \begin{equation} \|u\|_{p+1}\leq c_1\|Au\|_2,\quad \forall u\in D(A), \label{e2.2} \end{equation} \item[(A2)] There exist positive constants $m_0, n_0$, such that \begin{equation} M(z)\geq m_0,\quad N(z)\geq n_0,\quad \forall z\geq0.\label{e2.3} \end{equation} \item[(A3)] There exists positive constants $c_1,c_2$, such that \begin{equation} \|Au\|_2 \geq c_2\|A^{\frac{r}{2}}u\|_2\geq c_3\|u\|_2,\quad \forall u\in D(A), \; r\in(0,1]. \label{e2.4} \end{equation} \end{itemize} We denote the eigenvalues of the self-adjoint positive definite operator $A$ by $\{\lambda_j\}_{j\in N_+}$. Thus we have $0<\lambda_1<\lambda_2<\dots<\lambda_n<\dots$, and $\lambda_n\to+\infty$ $(n\to+\infty)$. The corresponding eigenvector series is $\{\omega_j\}_{j\in N_+}$. Let $D(A^s)=\{u\in D(A^{1/2}):A^su\in H\}$, and \begin{gather*} (u,v)_{D(A^s)}=(A^su,A^su) =\sum_{j=1}^{+\infty}\lambda_j^{2s}(u,\omega_j)(v,\omega_j), \quad \forall u,v\in D(A^s),\\ \|u\|_{D(A^s)}^2=(u,u)_{D(A^s)} =\sum_{j=1}^{+\infty}\lambda_j^{2s}(u,\omega_j)^2,~\forall u\in D(A^s). \end{gather*} Especially, $H=D(A^0)$, and we denote $V=D(A^\frac{\alpha}{2})$. Now, we state the following well posedness of the solution of system \eqref{e1.1}-\eqref{e1.2} which can be derived by Galerkin's method just as in \cite{ZJW}. \begin{lemma}\label{lem2.1} Assume that {\rm (A1)--(A3)} hold. If $(u_0,u_1),(v_0,v_1)\in V\cap D(A^{\beta})\times H$, then system \eqref{e1.1}-\eqref{e1.2} exists only one weak solution $(u,v)=(u(x,t),v(x,t))$ satisfying \begin{gather*} (u,v)\in L^{\infty}(0,T;D(A^{\beta}))\cap L^2(0,T;D(A^{\frac{\alpha+\beta}{2}})),\\ (u_t,v_t)\in L^{\infty}(0,T;H)\cap L^2(0,T;D(A^{\beta/2})). \end{gather*} \end{lemma} We let $$ \hat{M}(z)=\int_0^zM(s)ds,\quad \hat{N}(z)=\int_0^zN(s)ds, $$ and define the energy functional \begin{equation} E(t)=\frac{1}{2}\left(\|u_t\|_2^2+\|v_t\|_2^2\right)+J(t),\label{e2.5} \end{equation} where \begin{gather} J(t)=\frac{1}{2}w^2(t)-\int_\Omega F(u,v)dx, \nonumber \\ w(t)=\Big(\|Au\|_2^2+\|Av\|_2^2+\hat{M}(\|A^{\alpha/2}u\|_2^2 +\|A^{\alpha/2}v\|_2^2)\Big)^{1/2}.\label{e2.6} \end{gather} By a simple calculation, we obtain \begin{equation} E'(t)=-N(\|A^{\beta/2}u\|_2^2)\|A^{\beta/2}u_t\|_2^2 -N(\|A^{\beta/2}v\|_2^2)\|A^{\beta/2}v_t\|_2^2.\label{e2.7} \end{equation} From \eqref{e2.7} it follows that the energy $E(t)$ is non-increasing. Employing \eqref{e2.1}, \eqref{e2.2} and \eqref{e2.6}, we conclude that \begin{equation} \begin{aligned} \int_\Omega F(u,v)dx &\leq c_0\Big(\|u\|_{p+1}^{p+1}+\|v\|_{p+1}^{p+1}\Big)\\ &\leq c_0c_3^{p+1}\left(\|Au\|_2^{p+1}+\|Av\|_2^{p+1}\right)\\ &\leq 2c_0c_3^{p+1}w^{p+1}(t) := \frac{\eta}{p+1} w^{p+1}(t), \end{aligned} \label{e2.8} \end{equation} where $\eta=2(p+1)c_0c_3^{p+1}$ is a positive constant. \section{Exponential decay result} In this section, we prove a decay result for system \eqref{e1.1}-\eqref{e1.2}. For this purpose, we define the potential well $$ W=\Big\{(u,v)\in D(A)\times D(A):I(t)=w^2(t) -\gamma\int_\Omega F(u,v)dx>0\Big\}\cup(0,0). $$ \begin{lemma}\label{lem3.1} Let $(u,v)$ be the solution of system \eqref{e1.1}-\eqref{e1.2} and assume that {\rm (A1)--(A3)} hold. If $(u_0,v_0)\in W$, and $$ \zeta=\frac{\eta}{p+1}\Big(\frac{2\gamma}{\gamma-2}E(0)\Big)^{\frac{p-1}{2}} <\frac{1}{\gamma}, $$ then $$ \left(u(t),v(t)\right)\in W,\quad \forall t\geq0. $$ \end{lemma} \begin{proof} If $(u_0,v_0)\in W$, from the definition of $W$, we obtain $I(0)>0$. By the continuity of $I(t)$, there exists $T^*\in (0, \infty)$, such that for $t\in [0,T^*]$, $I(t)\geq 0$. Then we have $$ J(t)=\frac{\gamma-2}{2\gamma}w^2(t)+\frac{1}{\gamma}I(t) \geq\frac{\gamma-2}{2\gamma}w^2(t), \quad t\in[0,T^*], $$ thus we obtain $$ w^2(t)\leq\frac{2\gamma}{\gamma-2}J(t) \leq\frac{2\gamma}{\gamma-2}E(t)\leq\frac{2\gamma}{\gamma-2}E(0),\quad t\in[0,T^*]. $$ Combining this and \eqref{e2.8}, we obtain $$ \int_\Omega F(u,v)dx\leq\zeta w^2(t),\quad t\in[0,T^*]. $$ Then by the assumption on $\zeta$, we have $$ \left(u(t),v(t)\right)\in W,\quad t\in[0,T^*]. $$ Repeating the process, $T^*$ extends increasingly. \end{proof} \begin{lemma}\label{lem3.2} Let $(u,v)$ be the solution of system \eqref{e1.1}-\eqref{e1.2}, We assume that {\rm (A1)--(A3)} hold and the function $M(z)$ satisfies \begin{equation} \hat{M}(z)\leq M(z)z,\quad z\geq0.\label{e3.1} \end{equation} Then the functional $$ F(t)=(u,u_t)+(v,v_t)+\frac{1}{2}\hat{N}(\|A^{\beta/2}u\|_2^2) +\frac{1}{2}\hat{N}(\|A^{\beta/2}v\|_2^2) $$ satisfies \begin{equation} F'(t)\leq-I(t)+\|u_t\|_2^2+\|v_t\|_2^2. \label{e3.2} \end{equation} \end{lemma} \begin{proof} Differentiating $F(t)$ and by \eqref{e1.1}, we have \begin{align*} F'(t)&= \|u_t\|_2^2+\|v_t\|_2^2-M(\|A^{\alpha/2}u\|_2^2 +\|A^{\alpha/2}v\|_2^2)(\|A^{\alpha/2}u\|_2^2+\|A^{\alpha/2}v\|_2^2)\\ &\quad -\|Au\|_2^2-\|Av\|_2^2+\gamma\int_\Omega F(u,v)dx. \end{align*} By \eqref{e3.1} and the definition of $I(t)$, we obtain \eqref{e3.2}. \end{proof} We define the Lyapunov functional \begin{equation} L(t)=mE(t)+F(t),\label{e3.3} \end{equation} in which $m$ is a big positive constant to be determined later. \begin{theorem} \label{thm3.3} If the assumptions of Lemma \ref{lem3.1} and \eqref{e3.1} hold, and $N(z)\in L^{\infty}(0,\infty)$, then there exist two positive constants $\omega$ and $\kappa$, such that \begin{equation} E(t)\leq\kappa e^{-\omega t},\quad t\geq0.\label{e3.4} \end{equation} \end{theorem} \begin{proof} From \eqref{e2.4}, \eqref{e2.5}, \eqref{e3.3}, Young's inequality, Lemma \ref{lem3.1}, and combining with the condition $N(z)\in L^{\infty}(0,\infty)$, we have \begin{equation} \begin{aligned} &L(t)-\frac{m}{2}E(t) =F(t)+\frac{m}{2}E(t)\\ &\geq {-\frac{1}{2}\left(\|u\|_2^2+\|v\|_2^2+\|u_t\|_2^2+\|v_t\|_2^2\right) -c\|A^{\beta/2}u\|_2^2-c\|A^{\beta/2}v\|_2^2+\frac{m}{2}E(t)}\\ &\geq { -c\left(\|Au\|_2^2+\|Av\|_2^2+\|u_t\|_2^2+\|v_t\|_2^2\right) +\frac{m}{2}E(t)}\\ &={\left(\frac{m}{4}-c\right)\left(\|u_t\|_2^2+\|v_t\|_2^2\right) +\Big(\frac{m(\gamma-2)}{4\gamma}-c\Big)\left(\|Au\|_2^2 +\|Av\|_2^2\right)+\frac{m}{2\gamma}I(t)}\\ &\quad +\frac{m(\gamma-2)}{4\gamma}\hat{M}(\|A^{\alpha/2}u\|_2^2 +\|A^{\alpha/2}v\|_2^2)\\ &\geq {\left(\frac{m}{4}-c\right)\left(\|u_t\|_2^2+\|v_t\|_2^2\right) +\Big(\frac{m(\gamma-2)}{4\gamma}-c\Big)\left(\|Au\|_2^2+\|Av\|_2^2\right)}.\\ \end{aligned}\label{e3.5} \end{equation} On the other hand, by similar calculation, we obtain \begin{equation} \begin{aligned} &2mE(t)-L(t)={mE(t)-F(t)}\\ &\geq {\left(\frac{m}{2}-c\right)\left(\|u_t\|_2^2+\|v_t\|_2^2\right) +\Big(\frac{m(\gamma-2)}{2\gamma}-c\Big)\left(\|Au\|_2^2+\|Av\|_2^2\right)}. \end{aligned}\label{e3.6} \end{equation} We choose $N$ large enough, such that $$ L(t)-\frac{N}{2}E(t)\geq0,\quad 2NE(t)-L(t)\geq0, $$ thus we obtain \begin{equation} L(t)\sim E(t).\label{e3.7} \end{equation} From Lemma \ref{lem3.1}, we have a constant $\eta_1\in(0,1)$, such that $$ \gamma\int_\Omega F(u,v)dx\leq(1-\eta_1) w^2(t). $$ Hence we have $$ I(t)\geq \eta_1 w^2(t). $$ Then we arrive at \begin{equation} \begin{aligned} E(t)&={\frac{1}{2}\left(||u_t||_2^2+||v_t||_2^2\right) +\frac{\gamma-2}{2\gamma}w^2(t)+\frac{1}{\gamma}I(t)}\\ &\leq {\frac{1}{2}\left(||u_t||_2^2+||v_t||_2^2\right)+\eta_2 I(t)}, \end{aligned}\label{e3.8} \end{equation} where $\eta_2=\frac{\gamma-2}{2\gamma\eta_1}+\frac{1}{\gamma}$. Differentiating $L(t)$ and by \eqref{e2.3}, \eqref{e2.4}, \eqref{e2.7}, \eqref{e3.2}, we obtain $$ L'(t)\leq-(cN-1)(\|u_t\|_2^2+\|v_t\|_2^2)-I(t). $$ Let $N$ large enough, such that $cN-1>0$ and \eqref{e3.7} holds, exploiting \eqref{e3.8}, we have \[ L'(t)\leq-cE(t). \] Because of \eqref{e3.7}, we have some constant $\omega>0$ such that \begin{equation} L'(t)\leq-\omega L(t).\label{e3.9} \end{equation} Integrating \eqref{e3.9}, we have $L(t)\leq ce^{-\omega t}$. This completes the proof. \end{proof} \section{Blow-up result} Let $$ G(\lambda)=\frac{1}{2}\lambda^2-\frac{\eta}{p+1}\lambda^{p+1},\quad\lambda >0. $$ By calculation, we can get that $E_1:=G(\lambda_1)=\frac{p-1}{2(p+1)}\lambda_1^2$ is the maximum value of the function $G(\lambda)$, here $\lambda_1=\eta^{-\frac{1}{p-1}}$. \begin{lemma}\label{lem4.1} Let $(u,v)$ be the solution of system \eqref{e1.1}-\eqref{e1.2}. We assume that {\rm (A1),(A2)} hold, $w(0)>\lambda_1$ and $0\lambda_1,\quad t\geq 0, $$ and $$ \int_\Omega F(u,v)dx\geq\frac{\eta}{p+1}\lambda_2^{p+1}. $$ \end{lemma} \begin{lemma}[\cite{ZHH}] \label{lem4.2} Suppose that there is a positive, twice-differential function $Y(t)$ satisfies the inequality $$ Y''(t)Y(t)-\varsigma\left(Y'(t)\right)^2\geq 0,\quad t\geq 0, $$ where the constant $\varsigma>1$, then there is a $t^*<\frac{Y(0)}{(\varsigma-1)Y'(0)}$ such that $Y(t)\to\infty$ as $t\to t^*$. \end{lemma} \begin{theorem} \label{thm4.3} Let $(u,v)$ be the solution of system \eqref{e1.1}-\eqref{e1.2}. We assume that {\rm (A1), (A2)} hold and \begin{equation} \hat{M}(z)\geq M(z)z, \quad \hat{N}(z)\geq N(z)z.\label{e4.1} \end{equation} If anyone of the following conditions is satisfied: \begin{itemize} \item[(i)] $E(0)<0$; \item[(ii)] $E(0)=0$, $2(u_0,u_1)+2(v_0,v_1)>0$; \item[(iii)] $00$, \begin{align*} \Phi(t)&=\|u\|_2^2+\|v\|_2^2+\int^t_0\hat{N}(\|A^{\beta/2}u(t-s)\|_2^2)ds +\int^t_0\hat{N}(\|A^{\beta/2}v(t-s)\|_2^2)ds\\ &\quad +(T-t)\left[\hat{N}(\|A^{\beta/2}u_0\|_2^2)| +\hat{N}(\|A^{\beta/2}v_0\|_2^2)\right]+k_0(t+t_0)^2, \quad t\in[0,T], \end{align*} where $k_0$, $t_0\geq 0$ are constants to be determined later. Differentiating $\Phi(t)$, we have \begin{align*} \Phi'(t) &={2(u,u_t)+2(v,v_t)+2\int^t_0N(\|A^{\beta/2}u(s)\|_2^2) \left(A^{\beta/2}u_t(s),A^{\beta/2}u(s)\right)ds}\\ &\quad +2\int^t_0N(\|A^{\beta/2}v(s)\|_2^2)\left(A^{\beta/2}v_t(s), A^{\beta/2}v(s)\right)ds+2k_0(t+t_0). \end{align*} Taking the derivation of $\Phi'(t)$, we obtain \begin{align*} \Phi''(t) &={2\|u_t\|_2^2+2\|v_t\|_2^2+2\gamma\int_\Omega F(u,v)dx-2\|Au\|_2^2-2\|Av\|_2^2}\\ &\quad -2M(\|A^{\alpha/2}u\|_2^2+\|A^{\alpha/2}v\|_2^2)(\|A^{\alpha/2}u\|_2^2 +\|A^{\alpha/2}v\|_2^2)+2k_0. \end{align*} In the following, we deal with $\Phi''(t)$ in different situations. \smallskip \noindent\textbf{Cases $(i)$ and $(ii)$:} By \eqref{e2.5}, \eqref{e2.7}, \eqref{e4.1} and $\gamma>2$ we have \begin{align*} &\Phi''(t) \\ &=2\gamma\int^t_0N(\|A^{\beta/2}u(s)\|_2^2)\|A^{\beta/2}u_t(s)\|_2^2 +N(\|A^{\beta/2}v(s)\|_2^2)\|A^{\beta/2}v_t(s)\|_2^2ds \\ &\quad +2\gamma\left[E(t)-E(0)\right]+\Phi''(t) \\ &\geq (\gamma-2)\left[\|Au\|_2^2+\|Av\|_2^2+M(\|A^{\alpha/2}u\|_2^2 +\|A^{\alpha/2}v\|_2^2)(\|A^{\alpha/2}u\|_2^2+\|A^{\alpha/2}v\|_2^2)\right] \\ &\quad +2\gamma\int^t_0N(\|A^{\beta/2}u(s)\|_2^2)\|A^{\beta/2}u_t(s)\|_2^2 +N(\|A^{\beta/2}v(s)\|_2^2)\|A^{\beta/2}v_t(s)\|_2^2ds \\ &\quad +(\gamma+2)\left[\|u_t\|_2^2+\|v_t\|_2^2\right]-2\gamma E(0)+2k_0 \\ &\geq (\gamma+2)\Big[\int^t_0N(\|A^{\beta/2}u(s)\|_2^2) \|A^{\beta/2}u_t(s)\|_2^2+N(\|A^{\beta/2}v(s)\|_2^2)\|A^{\beta/2}v_t(s)\|_2^2ds \\ &\quad +\|u_t\|_2^2+\|v_t\|_2^2+k_0\Big]-\gamma\left[k_0+2E(0)\right]. \end{align*} Let \begin{gather*} P=\|u\|_2^2, \quad Q=\|v\|_2^2,\quad \tilde{P}=\|u_t\|_2^2,\quad \tilde{Q}=\|v_t\|_2^2,\\ R=\int^t_0N(\|A^{\beta/2}u(s)\|_2^2)\|A^{\beta/2}u(s)\|_2^2ds,\\ S=\int^t_0N(\|A^{\beta/2}v(s)\|_2^2)\|A^{\beta/2}v(s)\|_2^2ds,\\ \tilde{R}=\int^t_0N(\|A^{\beta/2}u(s)\|_2^2)\|A^{\beta/2}u_t(s)\|_2^2ds,\\ \tilde{S}=\int^t_0N(\|A^{\beta/2}v(s)\|_2^2)\|A^{\beta/2}v_t(s)\|_2^2ds. \end{gather*} We select $00. $$ Noticing that $\Phi(0)>0$, by Lemma \ref{lem4.2}, we conclude that there exist $t^*>0$, such that $$ \lim_{t\to t^*}\Phi(t)=\infty. $$ Since $t^*$ is independent of $T$, we assume that $t^*0$ and $\Phi'(0)>0$, then we use the same argument as Case (i). \smallskip \noindent\textbf{Case (iii):} By \eqref{e2.5}-\eqref{e2.7}, \eqref{e4.1}, $\gamma>3$ and Lemma \ref{lem4.1}, we obtain \begin{align*} &\Phi''(t) \\ &\geq \left(\gamma+1\right)\left(\|u_t\|_2^2+\|v_t\|_2^2\right) +(\gamma-3)w^2(t)+2k_0+2\int_\Omega F(u,v)dx-2(\gamma-1)E(t)\\ &= (\gamma+1)\left(\|u_t\|_2^2+\|v_t\|_2^2\right)+(\gamma-3)w^2(t)+2k_0 +2\int_\Omega F(u,v)dx-2(\gamma-1)E(0)\\ &\quad +2(\gamma-1)\int^t_0N(\|A^{\beta/2}u(s)\|_2^2)\|A^{\beta/2}u_t(s)\|_2^2 +N(\|A^{\beta/2}v(s)\|_2^2)\|A^{\beta/2}v_t(s)\|_2^2ds \\ &\geq \gamma\left[\tilde{P}+\tilde{Q}+\tilde{R}+\tilde{S}+k_0\right] +(\gamma-3)w(t)^2+2\int_\Omega F(u,v)dx \\ &\quad -2(\gamma-1)E(0)-(\gamma-2)k_0\\ &\geq \gamma\left[\tilde{P}+\tilde{Q}+\tilde{R}+\tilde{S}+k_0\right] +(\gamma-3)\lambda_2^2+\frac{2\eta}{p+1}\lambda_2^{p+1}\\ &-2(\gamma-1)E(0)-(\gamma-2)k_0\\ &\geq \gamma\left[\tilde{P}+\tilde{Q}+\tilde{R} +\tilde{S}+k_0\right]+\Big(\gamma-2-\frac{p-1}{p+1}\Big)\lambda_1^2\\ &\quad -2(\gamma-1)E(0)-(\gamma-2)k_0. \end{align*} By denoting $C:=\left(\gamma-2-\frac{p-1}{p+1}\right)\lambda_1^2-2(\gamma-1)E(0)$, we have $$ \Phi''(t)\geq\gamma\left[\tilde{P}+\tilde{Q}+\tilde{R}+\tilde{S} +k_0\right]+C-(\gamma-2)k_0. $$ Furthermore we know that $C>0$ because of $E(0)<\varrho E_1$. By selecting $$ 00. $$ Noticing that $\Phi(0)>0$, we repeat the process and conclude the desired result. \end{proof} \subsection*{Acknowledgements} The authors cordially thank the anonymous referee for his or her valuable comments and suggestions that lead to the improvement of this paper. This work was partially supported by NNSF of China (grant no. 61374089). \begin{thebibliography}{20} \bibitem{DS} P. D'Ancona, S. Spagnolo; \emph{A class of nonlinear hyperbolic problems with global solutions}, Arch. Rat. Mech. Anal. 124 (1993), 201-219. \bibitem{DS-1} P. D'Ancona, S. Spagnolo; \emph{Global solvability for the degenerate Kirchhoff equation with real analytic date}, Invent Math. 108 (1992), 247-262. \bibitem{EMM} Y. Ebihara, L. A. Medeiros, M. M. Miranda; \emph{Local solutions for a nonliear degenerare hyperbolic equation}, Nonlinear anal. TMA. 10 (1986), 27-40. \bibitem{ZJW} X. R. Hao, J. W. Zhang; \emph{The global solution for a class coupled of nonlinear abstract beam equations}, J. Dyn. Control, 13 (2015), 343-347 (in Chinese). \bibitem{I} R. Ikehata; \emph{On solutions to some quasilinear hyperbolic equations with nonlinear inhomogeneous terms}, Nonlinear Anal. TMA. 2 (1991), 181-203. \bibitem{K} G. Kirchhoff; \emph{Vorlesunden $\ddot{u}$ber Mechanik}, Teubner, Leipzig, 1883. \bibitem{LP} P. P. D. Lazo; \emph{Global solutions for a nonlinear wave equation}, Appl. Math. Compu. 200 (2008), 596-601. \bibitem{M} G. P. Menzala; \emph{On classical solutions of a quasilinear hyperbolic equation}, Nonlinear anal. TMA. 3 (1979), 613-627. \bibitem{M-1} T. Mizumachi; \emph{Time decay of solutions to degenerate Kirchhoff type equation}, Nonlinear Anal. TMA 3 (1998), 235-252. \bibitem{MM} C. L. Mu, J. Ma; \emph{On a system of nonlinear wave equations with Balakrishnan-Tayor damping}, Z. Angew. Math. Phys. 65 (2014), 91-113. \bibitem{R} J. E. M. Rivera; \emph{Smoothness effect and decay on a class of nonlinear evolution equation}, Ann. Fac. Sci. Toulouse. 1 (1992), 237-260. \bibitem{W} S. T. Wu; \emph{On decay and blow-up of soluions for a system of nonlinear wave equations}, J. Math. Anal. Appl. 394 (2012), 360-377. \bibitem{W-1} S. T. Wu; \emph{Blow-up of positive initial energy solutions for a system of nonlinear wave equations with supercritical sources}, J. Dyn. Control Syst. 20(2014), 207-227. \bibitem{ZDZ} J. W. Zhang, X. X. Ding, J. T. Zou; \emph{The global solution of coupled nonlinear equations}, Math. Prac. Theo. 13 (2011), 200-205. \bibitem{ZHH} H. W. Zhang, C. S. Hou, Q. Y. Hu; \emph{Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition}, Boundary Value Problems. 2013 (2013), 166,1-12. \end{thebibliography} \end{document}