\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 115, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/115\hfil Blow-up of solutions] {Blow-up of solutions for a nonlinear wave equation with nonnegative initial energy} \author[W. Liu, Y. Sun, G. Li\hfil EJDE-2013/115\hfilneg] {Wenjun Liu, Yun Sun, Gang Li} % in alphabetical order \address{Wenjun Liu \newline College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China} \email{wjliu@nuist.edu.cn} \address{Yun Sun \newline College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China} \email{shirlly\_@126.com} \address{Gang Li \newline College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China} \email{ligang@nuist.edu.cn} \thanks{Submitted January 18, 2013. Published May 6, 2013.} \subjclass[2000]{35B44, 35L10, 35L71} \keywords{Boundary damping term; interior source term; \hfill\break\indent nonlinear wave equation; nonnegative initial energy} \begin{abstract} In this article, we study a wave equation with nonlinear boundary damping and interior source term. We prove two blow-up results with nonnegative initial energy; thus we extend the blow-up results by Feng et al \cite{flz2012}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{s1} In this article, we study the following wave equation with nonlinear boundary damping and interior source term $$\begin{gathered} y_{tt}(x,t)-y_{xx}(x,t)=|y(x,t)|^{p-1}y(x,t), \quad (x,t)\in(0,L)\times(0,T),\\ y(0,t)=0,\quad y_{x}(L,t)=-|y_t(L,t)|^{m-1}y_t(L,t), \quad t\in[0,T),\\ y(x, 0)=y^{0}(x),\quad y_t(x, 0)=y^{1}(x), \quad x\in[0,L], \end{gathered} \label{1.1}$$ where $(0,L)$ is a bounded open interval in $\mathbb{R}$, $m>1$, $p>1$. The wave equation with interior damping term has been extensively studied and several results concerning existence, asymptotic behavior and blow-up have been established. When $m=1$, Levine \cite{l1974, l19742} proved that the solution blows up in finite time with negative initial energy. When $m>1$, Georgiev and Todorova \cite{gt1994} extended this result and established a global existence result if $m\geq p$ and a blow-up result if $mm$ and the initial data is inside an unstable set. For other wave equations with nonlinear source and damping terms, we can also refer the reader to \cite{cc2002,bl20081,bl2008, ccl2007,ly2011,l2012,t2009,w2009,yb2012,y2010} and references therein. Recently, Feng et al \cite{flz2012} considered \eqref{1.1} and obtained the blow-up results with one of the following conditions: (A) $2m\frac{4p}{(p-1)(p+1)}$. Later, Li et al \cite{lfw2012} studied the interaction between the interior damping $y_t(x,t)$ and the boundary source $|y(L,t)|^{p-1}y(L,t)+by(L,t)$ and established three sufficient conditions for the blow-up results with some necessary restriction on $b$ when the initial energy is positive or negative. Motivated by \cite{lfw2012}, we intend to extend the results in \cite{flz2012} with nonnegative initial energy. For this purpose, we use an improved relationship between $E_1$ and $\|y\|_{p+1}^{p+1}$ which is given in Lemma \ref{le3.1} below. This article is organized as follows. In Section \ref{s2}, we present some notation needed for our work and state our main results. In Section \ref{s3}, we give the proof of Theorem \ref{th2.2}. Section \ref{s4} is devoted to the proof Theorem \ref{th2.3}. \section{Notation and main results}\label{s2} We define the following functionals \begin{gather}\label{2.1} E(t)=\frac{1}{2}\|y_t(t)\|_2^2+\frac{1}{2}\|y_{x}(t)\|_2^2 -\frac{1}{p+1}\|y(t)\|_{p+1}^{p+1}, \\ \label{2.1'} I(t)=\|y_{x}(t)\|_2^2-\|y(t)\|_{p+1}^{p+1}, \end{gather} and as in \cite{flz2012} we introduce the notation: $\|\cdot\|_{q}=\|\cdot\|_{L^q(0,L)}$ and the Hilbert space $$\label{2.2} H_{\rm left}^1(0,L):=\{u\in H^1(0,L): u(0)=0\}.$$ Set $$\label{2.2'} E_1:=\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\alpha_0, \quad \alpha_0:=C_{*}^{-\frac{2(p+1)}{p-1}},$$ where $C_{*}$ is the optimal constant of the Sobolev embedding $\|y\|_{p+1}\leq C_{*}\|y_{x}\|_2$, for any $y\in H_{\rm left}^1(0,L)$. Next, we give a the existence of a local solution. \begin{theorem}[{\cite[Theorem 2.1]{flz2012}}] \label{th2.1} Assume that $(y^0, y^1)\in H_{\rm left}^1(0,L)\times L^2(0,L)$. Then \eqref{1.1} has a unique local solution $y(x,t)$ satisfying \begin{gather*} y(x,t)\in C(0,T_{m};H_{\rm left}^1(0,L)),\quad y_t(x,t)\in C(0,T_{m};L^2(0,L)), \\ y_t(L,t)\in L^{m+1}(0,T_{m}) \end{gather*} for some $T_{m}>0$, and the energy equality $$\label{2.4} E(t)+\int_0^{t}|y_t(L,\tau)|^{m+1}{\rm d}\tau=E(0)$$ holds for $0\leq t< T_{m}$. \end{theorem} Our main results are as follows. \begin{theorem}\label{th2.2} Let $y(x,t)$ be a solution of problem \eqref{1.1}. Assume that $2m\frac{4p+\frac{2p(p-1)}{p+1-\theta(p-1)}}{(p-1)(p+1) [1-\frac{2}{p+1-\theta(p-1)}]}, then the solution blows up in finite time. \end{theorem} \begin{remark} \rm When$E(0)<0$, the blow-up results have been proved in \cite{flz2012}. So, we consider here only the case$E(0)\geq0$. \end{remark} \begin{remark} \rm In the case$2m\geq p+1$, we note that the similar restriction on$L$as \eqref{2.5} has been used in \cite{flz2012}, which means that the larger the interval$(0, L)$is, the less the boundary damping effect. It is still the case when$I(0)<0$and$0\leq E(0)<\theta E_1$. \end{remark} \section{Proof of Theorem 2.2}\label{s3} In this section, we consider the blow-up result in the case$2m0. \end{lemma} \begin{proof} We adopt the manner which was first introduced in \cite{v1999}. From \eqref{2.1} and Sobolev embedding, we have $$E(t)\geq\frac{1}{2}\|y_{x}\|_2^2-\frac{1}{p+1}\|y\|_{p+1}^{p+1} \geq\frac{1}{2}\|y_{x}\|_2^2-\frac{C_{*}^{p+1}}{p+1}\|y_{x}\|_2^{p+1}.$$ Let $h(\xi)=\frac{1}{2}\xi-\frac{C_{*}^{p+1}}{p+1}\xi^{\frac{p+1}{2}}$, then $$E(t)\geq h(\xi)\quad \text{with } \xi=\|y_{x}\|_2^2.$$ It is easy to see that $h(\xi)$ is strictly increasing on $[0,\alpha_0)$, strictly decreasing on $(\alpha_0,+\infty)$ and takes its maximum value $E_1$ at $\alpha_0$. Since $I(0)<0$, we have $$\|y_{x}^0\|_2^2<\|y^0\|_{p+1}^{p+1}\leq C_{*}^{p+1}\|y_{x}^0\|_2^{p+1},$$ which leads to $$\|y_{x}^0\|_2^2>\alpha_0, \quad\text{for \alpha_0 defined by \eqref{2.2'}}.$$ Furthermore, since $$E_1>E(0)\geq E(t)\geq h(\|y_{x}\|_2^2), \quad \forall t\geq0,$$ there exists no time $t^*$ such that $\|y_{x}(t^*)\|_2^2=\alpha_0$. By the continuity of $\|y_{x}\|_2^2$, we obtain \begin{equation*} \|y_{x}\|_2^2>\alpha_0,\quad \forall t\geq0. \end{equation*} On the other hand, we have $$\frac{1}{p+1}\|y\|_{p+1}^{p+1}\geq -E(0)+\frac{1}{2}\|y_t\|_2^2 +\frac{1}{2}\|y_{x}\|_2^2>-\theta E_1+\frac{1}{2}\alpha_0 =\Big(\frac{p+1}{p-1}-\theta\Big)E_1,$$ which gives $$E_1<\frac{p-1}{2(p+1)} \frac{2}{(p+1)-\theta(p-1)}\|y\|_{p+1}^{p+1}.$$ Taking $\beta=\frac{2}{(p+1)-\theta(p-1)}\in(0,1)$, inequality \eqref{3.1} follows. \end{proof} Set $$H(t)=\theta E_1-E(t),$$ then it is clear that $H(t)$ is increasing, $H(t)\geq H(0)>0$ and $$\label{3.2} H(t)\leq \frac{\theta\beta(p-1)+2}{2(p+1)} \|y\|_{p+1}^{p+1}.$$ \begin{lemma} \label{le3.2} Under the assumptions of Lemma \ref{le3.1}, there exists a positive constant $C$ such that $$\label{3.3} \|y\|_{p+1}^{s}\leq C\|y\|_{p+1}^{p+1},$$ for any $2\leq s\leq p+1$. \end{lemma} \begin{proof} If $\|y\|_{p+1}^{p+1}\geq1$, then $\|y\|_{p+1}^{s}\leq C\|y\|_{p+1}^{p+1}$, since $s\leq p+1$. If $\|y\|_{p+1}^{p+1}<1$, then $\|y\|_{p+1}^{s}\leq \|y\|_{p+1}^{2}$, since $2\leq s$. Using the Sobolev embedding inequality, \eqref{2.1}, and Lemma \ref{le3.1}, we have $$\label{3.01} \|y\|_{p+1}^{2}\leq C_{*}\|y_{x}\|_2^{2}\leq2C_{*}\Big(E(t)+\|y\|_{p+1}^{p+1}\Big) \leq2C_{*}\Big(E_1+\|y\|_{p+1}^{p+1}\Big)\leq C\|y\|_{p+1}^{p+1}.$$ This completes the proof. \end{proof} As in \cite{flz2012}, we choose a constant $r$ such that \label{3.4} 0<\max \big\{\frac{2}{p+1}, \frac{m}{p+1-m}\big\}0to be chosen later, then from \eqref{3.12} and Lemma \ref{le3.4} we obtain \label{3.13} \begin{aligned} L'(t)&\geq \Big(1-\sigma-\frac{km\varepsilon}{m+1}\Big) H^{-\sigma}(t)|y_t(L,t)|^{m+1} +2\varepsilon\|y_t(t)\|_2^2 \\ &+\varepsilon\big[\frac{(p-1)(1-\theta\beta)}{p+1}-\frac{Ck^{-m}}{m+1}\big] \|y(t)\|_{p+1}^{p+1}. \end{aligned} Chooseklarge enough so that $$\frac{(p-1)(1-\theta\beta)}{p+1}-\frac{Ck^{-m}}{m+1}>0,$$ then \eqref{3.13} reduces to \begin{align*} L'(t)\geq\Big(1-\sigma-\frac{km\varepsilon}{m+1}\Big) H^{-\sigma}(t)|y_t(L,t)|^{m+1} +\varepsilon\gamma\big[\|y_t(t)\|_2^2+\|y(t)\|_{p+1}^{p+1}\big]. \end{align*} where\gamma>0$is the minimum of coefficients of$\|y_t(t)\|_2^2$and$\|y(t)\|_{p+1}^{p+1}$. We continue the remaining part as that of \cite[Theorem 2.2]{flz2012} to finish the proof. \end{proof} \section{Proof of Theorem 2.3}\label{s4} In this section, we consider the blow-up result in the case of$2m\geq p+1. Set \begin{align}\label{4.1} G(t)=E_1-E(t)+\epsilon\int_0^{L}xy_{x}(t)y_t(t){\rm d}x+\rho\epsilon\int_0^{L}y_t(t)y(t){\rm d}x, \end{align} with\rho\in \Big(\frac{2+\frac{\beta(p-1)}{2}}{(p-1)(1-\beta)}, \frac{L(p+1)}{2p}\Big)$, where$\beta$is given in the proof of Lemma \ref{le3.1} and$\epsilonis a small and positive constant satisfying \begin{align}\label{4.1'} G(0)=E_1-E(0)+\epsilon\int_0^{L}xy_{x}^{0}y^1{\rm d}x +\rho\epsilon\int_0^{L}y^1y^0{\rm d}x>0. \end{align} \begin{lemma}\label{le4.1} Under the assumptions of Theorem \ref{th2.3}, we haveG(t)>0$for all$t\geq0$. And there exists a positive constant$\eta>0such that $$\label{4.2} G'(t)\geq\eta[|y_t(L,t)|^{2m}+|y_t(L,t)|^{2}+|y_t(L,t)|^{p+1}].$$ \end{lemma} \begin{proof} As in \cite{flz2012}, using \eqref{1.1}, \eqref{2.1} and Lemma \ref{3.1}, we arrive at \label{4.3} \begin{aligned} G'(t) &\geq |y_t(L,t)|^{m+1}+\frac{L}{2}\epsilon|y_t(L,t)|^{2} +\frac{L}{2}\epsilon|y_t(L,t)|^{2m}+\frac{L\epsilon}{p+1}|y(L,t)|^{p+1} \\ &\quad -[\epsilon+2\rho\epsilon](E(t)-E_1)+2\rho\epsilon\|y_t(t)\|_2^2 -\rho\epsilon|y_t(L,t)|^{m}|y(L,t)| \\ &\quad +\epsilon[\frac{p-1}{p+1}\rho-\frac{2}{p+1}]\|y(t)\|_{p+1}^{p+1} -[\epsilon+2\rho\epsilon]E_1 \\ &\geq |y_t(L,t)|^{m+1}+\frac{L}{2}\epsilon|y_t(L,t)|^{2} +\frac{L}{2}\epsilon|y_t(L,t)|^{2m} +\frac{L\epsilon}{p+1}|y(L,t)|^{p+1}\\ &\quad +2\rho\epsilon\|y_t(t)\|_2^2 -\rho\epsilon|y_t(L,t)|^{m}|y(L,t)|\\ &\quad +\epsilon\big[\frac{p-1}{p+1}\rho-\frac{2}{p+1} -\frac{\beta(1+2\rho)(p-1)}{2(p+1)}\big]\|y(t)\|_{p+1}^{p+1}. \end{aligned} Using the choice of\rhoand Young's inequality, we obtain \label{4.4} \begin{aligned} G'(t) &\geq \frac{L}{2}\epsilon|y_t(L,t)|^{2} +\frac{L}{2}\epsilon|y_t(L,t)|^{2m} +\frac{L\epsilon}{p+1}|y(L,t)|^{p+1} \\ &\quad -\frac{p\rho\epsilon}{p+1}|y_t(L,t)|^{m\frac{p+1}{p}} -\frac{\epsilon\rho}{p+1}|y(L,t)|^{p+1}. \end{aligned} Then by repeating similar computations as that of \cite[Lemma 4.1]{flz2012}, we complete the proof. \end{proof} Set $$\label{4.5} F(t):=G^{1-\alpha}(t)+\mu\int_0^{L}y_t(t)y(t){\rm d}x\quad \text{with}\quad \alpha=\frac{p-1}{2(p+1)},$$ where\muis small enough to be chosen later. \begin{proof}[Proof of Theorem \ref{th2.3}] (Sketch) By repeating similar computations as that of \cite[Theorem 2.3]{flz2012}, from \eqref{3.1} we obtain %\label{4.6} \begin{align*} &F'(t)\\ &\geq (1-\alpha)G^{-\alpha}(t)\big(\eta-\mu CK^{\frac{1}{1-\alpha}}\big) [|y_t(L,t)|^{2m}+|y_t(L,t)|^{2}+|y(L,t)|^{p+1}] \\ &\quad +2\mu\|y_t\|_2^2-2\mu(E(t)-E_1)-2\mu E_1-\mu\alpha K^{-\frac{1}{\alpha}} G^{1-\alpha}(t)+\mu\frac{p-1}{p+1}\|y(t)\|_{p+1}^{p+1} \\ &\geq (1-\alpha)G^{-\alpha}(t)\big(\eta-\mu CK^{\frac{1}{1-\alpha}}\big) [|y_t(L,t)|^{2m}+|y_t(L,t)|^{2}+|y(L,t)|^{p+1}] \\ &\quad +2\mu\|y_t\|_2^2-\mu\alpha K^{-\frac{1}{\alpha}} G^{1-\alpha}(t)+\frac{\mu(p-1)(1-\beta)}{p+1}\|y(t)\|_{p+1}^{p+1}, \end{align*} whereK>0to be chosen later. Applying the Cauchy-Schwarz inequality, the Sobolev embedding and Lemma \ref{le3.1} to \eqref{4.1}, we obtain \label{4.7} \begin{aligned} G(t)&\leq E_1-E(t)+L\epsilon\int_0^{L}|y_{x}(t)||y_t(t)|{\rm d}x+\rho\epsilon\int_0^{L}|y_t(t)||y(t)|{\rm d}x \\ &\leq \Big(\frac{L\epsilon}{2}+\frac{\rho\epsilon}{2} -\frac{1}{2}\Big)\|y_t(t)\|_2^2+\Big(\frac{L\epsilon}{2} +\frac{\rho c_0^2\epsilon}{2}-\frac{1}{2}\Big)\|y_{x}(t)\|_2^2 \\ &\quad +\frac{2+\beta(p-1)}{2(p+1)}\|y(t)\|_{p+1}^{p+1}, \end{aligned} wherec_0$is the Sobolev embedding constant of$\|y\|_2\leq c_0\|y_{x}\|_2. From \eqref{2.1} and Lemma \ref{le3.1} it follows that \label{4.8} \begin{aligned} \|y_t(t)\|_2^2+\|y_{x}(t)\|_2^2 &=2E(t)+\frac{2}{p+1}\|y(t)\|_{p+1}^{p+1} \\ &<2E_1+\frac{2}{p+1}\|y(t)\|_{p+1}^{p+1}\\ &<\frac{2+\beta(p-1)}{p+1}\|y(t)\|_{p+1}^{p+1}. \end{aligned} Combining \eqref{4.7} and \eqref{4.8}, we obtain $$\label{4.9} G(t)\leq C\|y(t)\|_{p+1}^{p+1}.$$ Continuing as in the proof of \cite[Theorem 2.3]{flz2012} we can complete the proof. \end{proof} \subsection*{Acknowledgments} This work was partly supported by the Qing Lan Project of Jiangsu Province, by grant 41275009 from the National Natural Science Foundation of China, grant 11026211 from the the Tianyuan Fund of Mathematics, and grant CXLX12\_0490 from the JSPS Innovation Program. \begin{thebibliography}{00} \bibitem{cc2002} M. Aassila, M. M. Cavalcanti, V. N. Domingos Cavalcanti; \emph{Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term}, Calc. Var. 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