\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 20, pp. 1--7.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2013/20\hfil Blow-up solutions] {Blow-up solutions for a nonlinear wave equation with porous acoustic boundary conditions} \author[S.-T. Wu \hfil EJDE-2013/20\hfilneg] {Shun-Tang Wu} \address{Shun-Tang Wu \newline General Education Center, National Taipei University of Technology, Taipei, 106, Taiwan} \email{stwu@ntut.edu.tw} \thanks{Submitted August 28, 2012. Published January 23, 2013.} \subjclass[2000]{35L70, 35B40} \keywords{Blow-up; acoustic boundary conditions; exponential growth} \begin{abstract} We study a nonlinear wave equation with porous acoustic boundary conditions in a bounded domain. We prove a finite time blow-up for certain solutions with positive initial energy. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} We consider the following system of nonlinear wave equations with porous acoustic boundary conditions: \begin{gather} u_{tt}-\Delta u+\alpha (x)u+\phi (u_{t}) =j_1(u)\quad\text{in }\Omega \times [ 0,T), \label{e1.1} \\ u(x,t) = 0\quad\text{on }\Gamma_0\times (0,T), \label{e1.2} \\ u_{t}(x,t)+f(x)z_{t}+g(x)z = 0\quad\text{on }\Gamma_1\times (0,T), \label{e1.3} \\ \frac{\partial u}{\partial \nu }-h(x)z_{t}+\rho (u_{t}) = j_2(u)\quad \text{on }\Gamma_1\times (0,T), \label{e1.4} \\ u(x,0) = u_0(x) ,\text{ }u_{t}(x,0) =u_1(x) ,\quad x\in \Omega , \label{e1.5} \\ z(x,0) = z_0(x) ,\quad x\in \Gamma_1, \label{e1.6} \end{gather} where $\Omega$ is a bounded domain in $R^{n}$ $(n\geq 1)$ with a smooth boundary $\Gamma =\Gamma_0\cup \Gamma_1$. Here, $\Gamma_0$ and $\Gamma_1$ are closed and disjoint. Let $\nu$ be the unit normal vector pointing to the exterior of $\Omega$ and $\alpha :\Omega \to \mathbb{R}$, $f, g, h:\overline{\Gamma_1}\to \mathbb{R}$ and $j_1,j_2:\mathbb{R} \to \mathbb{R}$ are given functions. The system \eqref{e1.1}--\eqref{e1.6} is a model of nonlinear wave equations with acoustic boundary conditions which are described by \eqref{e1.3} and \eqref{e1.4}. These boundary conditions were introduced by Morse and Ingard \cite{m1} and developed by Beale and Rosencrans \cite{b1,b2,b3}. In recent years, questions related to wave equations with acoustic boundary conditions have been treated by many authors \cite{c1,f1,f2,m2,p1,r1,v1}. For example, Frota and Larkin \cite{f2} studied \eqref{e1.1}--\eqref{e1.6} with $\phi =\rho=j_1=j_2=0$ and they established the exponential decay result for suitably defined solutions. Recently, as $j_1=j_2=0$, Graber \cite{g1,g2} showed that the systems \eqref{e1.1}-\eqref{e1.6} generates a well-posed dynamical system by using semigroup theory. When one considers the presence of the double interaction between source and damping terms, both in the interior of $\Omega$ and on the boundary $\Gamma_1$, the analysis becomes more difficult. Very recently, Graber and Said-Houari \cite{g3} studied this challenging problem and obtained several results in local existence, global existence, the decay rate and blow-up results. Particularly, in the absence of boundary source, that is $j_2=0$, for certain initial data, the authors proved that the solution is unbounded and grows as an exponential function. However, the possibility of the solution that blows up in finite time is not addressed in that paper. Therefore, the intention of this paper is to investigate the blow-up phenomena of solutions for system \eqref{e1.1}-\eqref{e1.6} without imposing the boundary source. In this way, we can extend this unbounded result of \cite{g3} to a blow-up result with positive initial energy. The content of this paper is organized as follows. In section 2, we state the local existence result and the energy identity which is crucial in establishing the blow-up result in finite time. In section 3, we study the blow-up problem for the initial energy being positive. \section{Preliminaries} In this section, we present some material which will be used throughout this work. First, we introduce the set \begin{equation*} H_{\Gamma_0}^1=\{ u\in H^1(\Omega ) :u| _{\Gamma_0}=0 \} , \end{equation*} and endow $H_{\Gamma_0}^1$ with the Hilbert structure induced by $H^1(\Omega )$, we have that $H_{\Gamma_0}^1$ is a Hilbert space. For simplicity, we denote $\| \cdot \| _{p}=\| \cdot \|_{L^p(\Omega ) }$, $\| \cdot \|_{p,\Gamma }=\| \cdot \|_{L^p(\Gamma ) }$, $1\leq p\leq \infty$, $\| u\|_{\alpha }^2=\| \nabla u\| _2^2+\int_{\Omega }\alpha (x)u^2(x)dx$ and $\| u\|_{gh}^2=\int_{\Gamma_1}g(x)h(x)u^2(x)d\Gamma$. The following assumptions for problem \eqref{e1.1}-\eqref{e1.6} were used in \cite{g3}. \begin{itemize} \item[(A1)] The functions $j_1(s)=|s| ^{p-1}s$ and $j_2(s)=0$, where $p\geq 1$ is such that $H_{\Gamma_0}^1(\Omega ) \hookrightarrow L^{p+1}( \Omega )$. \item[(A2)] $\phi , \rho :\mathbb{R}\to \mathbb{R}$ are continuous and increasing functions with $\phi (0)=\rho (0)=0$. In addition, there exist positive constants $a_i$ and $b_i$ $i=r,q$ such that \begin{gather} a_r| s| ^{r+1} \leq \phi (s)s\leq b_r|s| ^{r+1},\; r\geq 1, \label{e2.1} \\ a_q| s| ^{q+1} \leq \rho (s)s\leq b_q|s| ^{q+1},\; q\geq 1. \label{e2.2} \end{gather} \item[(A3)] The functions $\alpha ,f, g, h$ are essentially bounded such that $f>0$, $g>0$, $h>0$ and $\alpha \geq 0$. (If $\alpha =0$, $\Gamma_0$ is assumed to have a non-empty interior such that the Poincar\`e inequality is applicable.) \end{itemize} Next, the energy function associated with problem \eqref{e1.1}--\eqref{e1.6}, with $j_2=0$, is defined as $$E(t)=\frac{1}{2}\| u_{t}\|_2^2+\frac{1}{2}\Big( \| u\|_{\alpha }^2+\| z\| _{gh}^2\Big) -\frac{1}{p+1}\| u\|_{p+1}^{p+1}. \label{e2.3}$$ Then, we are ready to state the following local existence result and energy identity. \begin{lemma}[\cite{g3}] \label{lem2.1} Suppose that {\rm (A1)--(A3)} hold, and that $u_0\in H_{\Gamma_0}^1(\Omega )$, $u_1\in L^2(\Omega )$ and $z_0\in L^2(\Gamma_1)$. Then the system {\rm \eqref{e1.1}--\eqref{e1.6}} with $j_2=0$ admits a unique solution $(u,z)$ such that, for $T>0$, $u\in C([0,\text{ }T);H_{\Gamma_0}^1(\Omega ) ) \cap C^1([0,\text{ }T);L^2(\Omega ) ) ,\quad z\in C([0,T);L^2(\Gamma_1)) .$ Moreover, the energy satisfies $$E(0)=E(T)+\int_0^{T}\int_{\Omega }\phi (u_{t})u_{t}dxdt+\int_0^{T}\int_{\Gamma_1}(\rho (u_{t})u_{t}+fhz_{t}^2) d\Gamma dt. \label{e2.4}$$ \end{lemma} Note that \eqref{e2.4} shows that the energy is a non-increasing function along trajectories. \section{Blow-up of Solutions} In this section, we state and prove our main result. First, we define a functional $G$ which helps in establishing desired results. Let $$G(x)= \frac{1}{2}x^2-\frac{B_1^{p+1}}{p+1}x^{p+1}, \quad x>0,$$ where $B_1^{-1}=\inf \{ \| \nabla u\|_2: u\in H_{\Gamma_0}^1(\Omega ) ,\; \| u\|_{p+1}=1\}$. Then, $G$ has a maximum at $\lambda_1=B_1^{-\frac{ p+1}{p-1}}$ with the maximum value $E_1\equiv G(\lambda_1)=(\frac{1}{2}-\frac{1}{p+1}) \lambda_1^2.$ The next Lemma will play an important role in proving our result. \begin{lemma}[\cite{g3}] \label{lem3.1} Suppose that {\rm (A1)--(A3)} hold, and that $u_0\in H_{\Gamma_0}^1(\Omega )$, $u_1\in L^2(\Omega )$ and $z_0\in L^2(\Gamma_1)$. Let $(u,z)$ be a solution of \eqref{e1.1}--\eqref{e1.6} with $j_2=0$. Assume that $E(0)\lambda_1$. Then there exists $\lambda_2>\lambda_1$ such that, for all $t\geq 0$, \begin{gather} (\| u\|_{\alpha }^2+\| z\|_{gh}^2) ^{1/2}\geq \lambda_2, \label{e3.1} \\ \| u\|_{p+1}\geq B_1\lambda_2. \label{e3.2} \end{gather} \end{lemma} Now, we are ready to state and prove our main result. Our proof technique follows the arguments of \cite{g3} and some estimates obtained in \cite{g4}. \begin{theorem} \label{thm3.2} Suppose that {\rm (A1)--(A3)} hold, and that $u_0\in H_{\Gamma_0}^1(\Omega )$, $u_1\in L^2(\Omega )$ $z_0\in L^2(\Gamma_1)$. Assume further that $p>\max (r,2q-1)$. Then any solution of \eqref{e1.1}-\eqref{e1.6} with $j_2=0$and satisfying $E(0)\lambda_1$ blows up at a finite time. \end{theorem} \begin{proof} We suppose that the solution exists for all time and we reach to a contradiction. To achieve this, we set $$H(t) =E_1-E(t) ,\quad t\geq 0. \label{e3.3}$$ Then, by \eqref{e2.4}, we see that $H'(t)\geq 0$. From \eqref{e3.1}, the definition of $E(t)$ and $E_1=(\frac{1}{2}-\frac{1}{p+1})\lambda_1^2$, we deduce that, for all $t\geq 0$, 0\frac{p-1}{p+1}-2E_1(B_1^{p+1}\lambda_1^{p+1})^{-1}=0. \end{equation*} By \eqref{e2.1}, H\"{o}lder inequality and Young's inequality, we see that, for $\delta_1>0$, $$\big| \int_{\Omega }u\phi (u_{t})dx\big| \leq \frac{b_r\delta _1^{r+1}}{r+1}\| u\|_{r+1}^{r+1}+\frac{b_rr\delta _1^{-\frac{r+1}{r}}}{r+1}\| u_{t}\|_{r+1}^{r+1}, \label{e3.10}$$ A substitution of \eqref{e3.10} into \eqref{e3.9} leads to \begin{aligned} A'(t) &\geq (1-\sigma )H(t)^{-\sigma }H'(t) +2\varepsilon \| u_{t}\|_2^2+\varepsilon c_1\| u\|_{p+1}^{p+1}\\ &\quad +2\varepsilon \| z\|_{gh}^2 -\varepsilon \int_{\Gamma_1}u\rho (u_{t})d\Gamma \\ &\quad-\varepsilon \Big(\frac{b_r\delta_1^{r+1}}{r+1}\| u\|_{r+1}^{r+1}+\frac{b_rr\delta_1^{-\frac{r+1}{r}}}{r+1} \| u_{t}\|_{r+1}^{r+1}\Big) +2\varepsilon H(t), \end{aligned}\label{e3.11} At this point, for a large positive constant $M_1$ to be chosen later, picking $\delta_1$ such that $\delta_1^{-\frac{r+1}{r}}=M_1H(t)^{-\sigma }$ and using the fact $$H'(t)\geq a_r\| u_{t}\| _{r+1}^{r+1}+a_q\| u_{t}\|_{q+1,\Gamma}^{q+1} +\int_{\Gamma_1}fhz_{t}^2d\Gamma \label{e3.12}$$ by \eqref{e2.4} and (A1) we have \begin{aligned} A'(t) &\geq (1-\sigma -\frac{\varepsilon rb_rM_1}{ a_r(r+1) })H(t)^{-\sigma }H'(t)+2\varepsilon \| u_{t}\|_2^2+\varepsilon c_1\| u\|_{p+1}^{p+1}\\ &\quad +2\varepsilon \| z\|_{gh}^2 -\varepsilon \int_{\Gamma_1}u\rho (u_{t})d\Gamma -\frac{\varepsilon b_rM_1^{-r}}{r+1}H(t)^{\sigma r}\| u\| _{r+1}^{r+1}+2\varepsilon H(t). \end{aligned} \label{e3.13} In addition, using \eqref{e3.4} and the inequality \begin{equation*} \chi ^{\gamma }\leq \chi +1\leq (1+\frac{1}{\omega }) ( \chi +\omega ) ,\quad \forall \chi \geq 0,\;0<\gamma \leq 1, \;\omega >0, \end{equation*} with $\chi =\frac{1}{p+1}\| u\|_{p+1}^{p+1}$ and $\omega =H(0)$ and noting that $p>r$, and $0<\sigma r+\frac{r+1}{p+1}\leq 1$ by \eqref{e3.6}, we have \begin{aligned} H(t)^{\sigma r}\| u\|_{r+1}^{r+1} &\leq c_2H(t)^{\sigma r}\Big(\| u\|_{p+1}^{p+1}\Big) ^{\frac{r+1}{p+1}}\\ &\leq c_3\Big(\frac{1}{p+1}\| u\|_{p+1}^{p+1}\Big) ^{\sigma r+\frac{r+1}{p+1}} \\ &\leq c_3d\Big(\frac{1}{p+1}\| u\|_{p+1}^{p+1}+H(t)\Big) , \end{aligned} \label{e3.14} where $c_2=\operatorname{vol}(\Omega ) ^{\frac{p-r}{p+1}}$ and $c_3=(p+1) ^{\frac{r+1}{p+1}}\cdot c_2$ and $d=1+\frac{1}{H(0)}$. Combining \eqref{e3.14} with \eqref{e3.13}, we obtain \begin{aligned} A'(t) &\geq (1-\sigma -\frac{\varepsilon rb_rM_1}{ a_r(r+1) })H(t)^{-\sigma }H'(t)+2\varepsilon \| u_{t}\|_2^2+\varepsilon (c_1-c_4) \| u\|_{p+1}^{p+1} \\ &\quad +2\varepsilon \| z\|_{gh}^2+\varepsilon ( 2-(p+1)c_4) H(t)-\varepsilon \int_{\Gamma_1}u\rho (u_{t})d\Gamma . \end{aligned} \label{e3.15} with $c_4=\frac{b_rc_3dM_1^{-r}}{(p+1)(r+1)}$. Next, we will follow the arguments as in \cite{g4} to estimate the last term on the right hand side of \eqref{e3.15}. For this purpose, let us recall the following trace and interpolation theorems \cite{a1,r2} $$\| u\|_{q+1,\Gamma }\leq C\| u\|_{W^{s,q+1}}, \label{e3.16}$$ which holds for some positive constant $C$, $q\geq 0$, $0\frac{1}{q+1}$. $$W^{1-\theta ,\tau }(\Omega ) =[ H^1(\Omega ) ,L^{p+1}(\Omega ) ]_{\theta }, \label{e3.17}$$ where $\frac{1}{\tau }=\frac{1-\theta }{2}+\frac{\theta }{p+1}$, $\theta \in[ 0,1]$ and $[\cdot ,\cdot ]_{\theta }$ denotes the interpolation bracket. We note from $q\geq 1$ and $p>2q-1$ that $\frac{1}{q+1}\leq \frac{p-1}{2(p-q)}<1$. Then, we choose $\beta$ satisfying $$\frac{p-1}{2(p-q)}\leq \beta <1 \label{e3.18}$$ and select $\theta$ such that \begin{equation*} 1-\theta =\frac{1}{\beta (q+1) }, \quad \tau =\frac{2(p+1)}{(1-\theta )(p+1)+2\theta }, \end{equation*} which imply that $1-\theta >\frac{1}{q+1}$ and $\tau \geq q+1$. From \eqref{e3.16}, \eqref{e3.17} and Young's inequality, we have \begin{align*} \| u\|_{q+1,\Gamma } &\leq C\| u\| _{W^{1-\theta ,q+1}(\Omega ) } \leq C\| u\| _{W^{1-\theta ,\tau }(\Omega ) } \leq C\| u\|_{\alpha }^{1-\theta }\| u\|_{p+1}^{\theta } \\ &= C\| u\|_{\alpha }^{\frac{1}{\beta (q+1) } }\| u\|_{p+1}^{1-\frac{1}{\beta (q+1) }}\\ &\leq C\Big(\| u\|_{\alpha }^{\frac{2\beta }{q+1}}+\| u\|_{p+1}^{\frac{2\beta ^2(q+1)-2\beta }{(2\beta ^2-1) (q+1) }}\Big) , \end{align*} where $C$ is a generic positive constant. Further, as in \cite{g4}, there exists $\beta$ satisfying \eqref{e3.18} such that $\frac{2\beta ^2(q+1)-2\beta }{(2\beta ^2-1) (q+1) } =\frac{(p+1) \beta }{q+1}$. Thus, $$\| u\|_{q+1,\Gamma }\leq C\Big(\| u\| _{\alpha }^{\frac{2\beta }{q+1}}+\| u\|_{p+1}^{\frac{ (p+1) \beta }{q+1}}\Big) . \label{e3.19}$$ Besides, we observe from the definition of $E(t)$ by \eqref{e2.3} and $E(0)0$, \begin{align*} &| \int_{\Gamma_1}u\rho (u_{t})d\Gamma |\\ & \leq c_6\Big(E_1+\frac{1}{p+1}\| u\|_{p+1}^{p+1}\Big) ^{ \frac{\beta -1}{q+1}}\Big(E_1+\frac{1}{p+1}\| u\| _{p+1}^{p+1}\Big) ^{\frac{1}{q+1}}\| u_{t}\|_{q+1,\Gamma }^{q} \\ &\leq c_6\Big(E_1+\frac{1}{p+1}\| u\| _{p+1}^{p+1}\Big) ^{\frac{\beta -1}{q+1}} \Big[\delta \Big(E_1+\frac{1 }{p+1}\| u\|_{p+1}^{p+1}\Big) +c_{\delta }\| u_{t}\|_{q+1,\Gamma }^{q+1}\Big] \\ &\leq c_6\delta H(0)^{\frac{\beta -1}{q+1}}\Big(E_1+\frac{1}{p+1} \| u\|_{p+1}^{p+1}\Big) +c_{\delta }c_6a_q^{-1}H(0)^{ \frac{\beta -1}{q+1}+\sigma }H(t)^{-\sigma }H'(t). \end{align*} Thus, \eqref{e3.15} becomes \begin{align*} A'(t) &\geq (1-\sigma -\varepsilon c_7) H(t)^{-\sigma }H'(t)+2\varepsilon \| u_{t}\|_2^2\\ &\quad +\varepsilon \Big(c_1-c_4-\frac{c_6\delta H(0)^{\frac{\beta -1}{q+1}}}{p+1}\Big) \| u\|_{p+1}^{p+1} \\ &\quad +2\varepsilon \| z\|_{gh}^2+\varepsilon ( 2-(p+1)c_4) H(t)-c_6\delta \varepsilon H(0)^{\frac{\beta -1}{q+1}}E_1, \end{align*} where $c_7=\frac{rb_rM_1}{a_r(r+1) } +c_{\delta}c_6a_q^{-1}H(0)^{\frac{\beta -1}{q+1}+\sigma }$. Employing the estimate \eqref{e3.8} again, we arrive at \begin{align*} A'(t) &\geq (1-\sigma -\varepsilon c_7) H(t)^{-\sigma }H'(t)+2\varepsilon \| u_{t}\| _2^2+\varepsilon (c_1-c_4-\delta c_{8}) \| u\|_{p+1}^{p+1} \\ &\quad +2\varepsilon \| z\|_{gh}^2+\varepsilon ( 2-(p+1)c_4) H(t), \end{align*} where $c_{8}=c_6H(0)^{\frac{\beta -1}{q+1}}(\frac{1}{p+1} +E_1(B_1^{p+1}\lambda_2^{p+1}) ^{-1}) .$ Now, we choose $M_1$ large enough such that $2-(p+1)c_4>0$ and $c_1-c_4> \frac{c_1}{2}$. Once $M_1$ is fixed, we select $\delta$ small enough such that $\frac{c_1}{2}-\delta c_{8}>0$. Then, pick $\varepsilon$ small enough such that $1-\sigma -\varepsilon c_7\geq 0$ and $A(0)>0$. Thus, there exists $K>0$ such that $$\label{e3.20} \begin{gathered} A'(t) \geq \varepsilon K(\|u_{t}\|_2^2+\| u\|_{p+1}^{p+1}+H(t) +\| z\|_{gh}^2) , \\ A(t)\geq A(0)>0,\quad \text{for } t\geq 0. \end{gathered}$$ On the other hand, from the result of Graber et al \cite[Lemma 6.5]{g3}, we have $$A(t) ^{\frac{1}{1-\sigma }} \leq c_{9}\Big(\| u_{t}\|_2^2+\| u\|_{p+1}^{p+1}+H(t) +\| z\|_{gh}^2\Big) ,\quad t\geq 0. \label{e3.21}$$ Combining \eqref{e3.21} with \eqref{e3.20}, we obtain $$\ A'(t) \geq c_{10}A(t) ^{\frac{1}{1-\sigma }},\ t\geq 0, \label{e3.22}$$ where $c_i$, $i=9,10$, are positive constants. Thus, inequality \eqref{e3.22} leads to a blow-up result in a finite time $T$ with \begin{equation*} 0