\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 76, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/76\hfil Blowup and asymptotic stability] {Blowup and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms} \author[Q. Y. Hu, H. W. Zhang\hfil EJDE-2007/76\hfilneg] {Qingying Hu, Hongwei Zhang} % in alphabetical order \thanks{Submitted February 27, 2007. Published May 22, 2007.} \subjclass[2000]{35B40} \keywords{Wave equation; degenerate damping and source terms; \hfill\break\indent asymptotic stability; blow up of solutions} \begin{abstract} This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation $$u_{tt}-\Delta u +|u|^kj'(u_t)=|u|^{p-1}u \quad \text{in }\Omega \times (0,T),$$ where $p>1$ and $j'$ denotes the derivative of a $C^1$ convex and real value function $j$. We prove that every weak solution is asymptotically stability, for every $m$ such that $0k+m$ and the initial data is positive, but appropriately bounded. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction} In this article we study the initial boundary value problem \begin{gather} u_{tt}-\Delta u+|u|^kj'(u_t)=|u|^{p-1}u, \quad\text{in }\Omega\times (0,T), \label{1.1}\\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad\text{in }\Omega,\label{1.2}\\ u(x,t)=0, \quad\text{on }\Gamma\times (0,T), \label{1.3} \end{gather} where $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\Gamma$ and $j(s)$ is a $C^1$ convex real function defined on $R$, and $j'$ denotes the derivative of $j$ \cite{BLR1}. Furthermore, the following assumptions on the convex function $j$ and the parameters $k,m,p$ are imposed throughout the paper. \subsection*{Assumptions} \quad \begin{itemize} \item[(A1)] $k,m,p>0$, and $k<\frac {n}{n-2}$, $p+1<\frac {2n}{n-2}$ if $n \ge 3$; \item[(A2)] There exist positive constants $C,C_0,C_1$ such that for all $s,v \in R$, $j(s) \ge C|s|^{m+1}$, $|j'(s)|\le C_0|s|^m$, $(j'(s)-j'(v))(s-v)\ge C_1|s-v|^{m+1}$. \end{itemize} The partial differential equation \eqref{1.1} is a special case of the prototype evolution equation $$\label{1.4} u_{tt}-\Delta u+Q(x,t,u,u_t)=f(x,u),$$ where the nonlinearities satisfy the structural conditions $vQ(x,t,u,v)\geq0$, $$Q(x,t,u,0)=f(x,0)=0$$ and $f(x,u)\sim |u|^{p-1}u$ for large $|u|$. Various special cases of \eqref{1.4} arise in many contexts, for instance, in classical mechanics, fluid dynamics, quantum field theory, see \cite{Jo} and \cite{Se}. A special case of \eqref{1.1}, is the following well known polynomially damped wave equation studied extensively in the literature(see for instance \cite{PR,RS}), $$\label{1.5} u_{tt}-\Delta u+|u|^k|u_t|^{m-1}u_t=|u|^{p-1}u.$$ Indeed, by taking $j(s)=\frac {1}{m+1}|s|^{m+1}$ we easily verify that Assumption (A1) and (A2) is satisfied. It is easy to see in this case that equation \eqref{1.1} is equivalent to \eqref{1.5}. It is worth noting that there has been an extensive body of work on the global existence and nonexistence for the equation \eqref{1.1} with $k=0$, see, for example \cite{BV}-\cite{LRS}, \cite{PS1}-\cite{PuSe3},\cite{Se,BLR2} and the references therein. One of the pioneering papers in this area was by Lions and Strauss \cite{LS}. We also note here the work of Georgiev and Todorova \cite{GT} and Levine and Serrin \cite{LeSe}. The situation, however, is different when the damping is degenerate. From the applications point of view degenerate problem of this type arise quite often in specific physical contexts: for example when the friction is modulated by the strain. However, from the mathematical point of view this leads to that some standard arguments to establish the existence of solutions to problem \eqref{1.1}-\eqref{1.3} is not applicable. These difficulties makes the problem interesting and the analysis more subtle. The problem with degenerate damping has been first addressed in Levine and Serrin \cite{LeSe}, where the global nonexistence of solutions was shown for the case $k+mm+k$ and the initial energy is negative. The negativity of the initial energy was used to prove blow up in the above paper \cite{BLR2,BLR3,PR,RS}. However, the blow up of the solutions for \eqref{1.1}-\eqref{1.3} in case of positive initial energy has not been discussed, and the asymptotic behavior of the solutions for \eqref{1.1}-\eqref{1.3} is much less understood. In this paper, following the ideas of potential well" theory introduced by Payne and Sattinger \cite{PS1}, we extend the results about asymptotic stability and blowup of the solution to \eqref{1.1}-\eqref{1.3} with $k=0$ (see, for example, \cite{BV,PS2,Vi1,Vi2,ZC} to the problem \eqref{1.1}-\eqref{1.3} with $k>0$. It is worth mentioning here that Levine, Park and Serrin [9] studied the existence and nonexistence of the solution to the quasilinear evolution equation of formally parabolic type, namely $$\label{1.6} Q(t,u,u_t)+A(t,u)=f(t,u).$$ The purpose of the paper is, first, to show that the weak solution of the problem \eqref{1.1}-\eqref{1.3} blow up in the case of positive initial energy $E(0)>0$ and $p>k+m$, which we do in section 3. The another purpose of this paper is to give an asymptotic stability results of the problem \eqref{1.1}-\eqref{1.3} with $00$ such that the initial boundary problem \eqref{1.1}-\eqref{1.3} has a unique weak solution on $[0,T]$ if $p\le k+m$. \end{theorem} Now, we define the energy associated with problem \eqref{1.1}-\eqref{1.3} by $$E(t)=\frac{1}{2}\|u_t(t)\|^2+\frac{1}{2}\|\nabla u(t)\|^2-\frac{1}{p+1}\|u(t)\|_{p+1}^{p+1}.$$ We see that the energy has the so-called energy identity $$E(t)+\int_0^t\int_{\Omega}|u(s)|^kj(u_t)(s)ds=E(0),$$ where $$E(0)=\frac{1}{2}\|u_1\|^2+\frac{1}{2}\|\nabla u_0\|^2-\frac{1}{p+1}\|u_0\|_{p+1}^{p+1}.$$ It is clear that $$E'(t)=-\int_{\Omega}|u(s)|^kj(u_t)(s)ds\leq 0 \label{2.1}$$ and $E(t)$ is a non-increasing function in time, then $$E(t)\geq E(0).\label{2.2}$$ Finally, we set \begin{gather*} \lambda_1=B_1^{-\frac{2}{p-1}},\quad E_1=(\frac{1}{2}-\frac{1}{p+1})\lambda_1^{p+1}, \\ \lambda_2=(\frac{1}{(p+1)B_1^2})^{\frac{1}{p-1}},\quad E_2=\frac{p+1}{2}(\frac{1}{2}-\frac{1}{p+1})\lambda_2^{p+1}, \\ {\sum }_{1}=\{(\lambda,E)\in R^2,\lambda>\lambda_1,0k+m$and$u$be a weak solution to \eqref{1.1}-\eqref{1.3} on the interval$[0,T]$in the sense of Definition \ref{def2.1}. \begin{lemma} \label{lem3.1} Let$(\|u_0\|_{p+1},E(0))\in \sum_1$, then$E(t)\leq E_0$for all$t\in [0,T]$, and there exist$\lambda_0>\lambda_1$such that$\|u(t)\|_{p+1}\geq\lambda_0>\lambda_1$for all$t\in [0,T]$. \end{lemma} The proof is similar to that of \cite[Lemma 1]{Vi1}, so we omit it. \begin{theorem} \label{thm3.2} Let$(\|u_0\|_{p+1},E(0))\in \sum_1,~p>k+m$, and$u$be a weak solution to \eqref{1.1}-\eqref{1.3} on the interval$[0,T]$in the sense of Definition \ref{def2.1}, then$T$is necessarily finite, i.e.$u$can not be continued for all$t>0$. \end{theorem} \begin{proof} We argue by contradiction. Let$F(t)=\|u(t)\|^2$,$H(t)=E_1-E(t)$. From \eqref{2.1}, we have $$H'(t)=-E'(t)=\int_{\Omega}|u(t)|^kj(u_t)(t)dx\geq 0.\label{3.1}$$ Therefore,$H(t)$is an increasing function, then $$\label{3.2} H(t)\geq H(0)=E_1-E(0)>0, \quad t\geq 0.$$ Next, by the definition of$E(t)and Lemma \ref{lem3.1}, \begin{aligned} H(t)&\leq E_1-\frac{1}{2}\|\nabla u(t)\|^2+\frac{1}{p+1}\|u(t)\|_{p+1}^{p+1}\\ &\leq E_1-\frac{1}{2}B^{-2}_1\lambda_1^2+\frac{1}{p+1}\|u(t)\|_{p+1}^{p+1}, \quad t\geq 0. \end{aligned}\label{3.3} Hence, sinceE_1-\frac{1}{2}B^{-2}_1\lambda_1^2= -\frac{1}{p+1}\lambda_1^{p+1}<0$, we have \label{3.4} 01$, or $E_1\|u(t)\|_{p+1}^{p+1}\lambda_0^{-(p+1)}>E_1$), \begin{aligned} \frac{1}{2}F''(t)&\geq 2\|u_t(t)\|^2+(1-\frac{2}{p+1}-2E_1\lambda_0^{-(p+1)})\|u(t)\|_{p+1}^{p+1}+2H(t)-I(t)\\ &=2\|u_t(t)\|^2+C_2\|u(t)\|_{p+1}^{p+1}+2H(t)-I(t), \end{aligned}\label{3.5} where $C_2=1-\frac{2}{p+1}-2E_1\lambda_0^{-(p+1)}>0$, because $\lambda_0>\lambda_1$ by Lemma \ref{lem3.1}. Now, to estimate the last term $I(t)$ in \eqref{3.5}, since $p>k+m$ and Assumption (A1) and (A2) and by applying Holder's inequality and Young's inequality, we obtain \begin{aligned} |I(t)|&\leq C_0\int_{\Omega}|u(t)|^{k+1-\frac{k+m+1}{m+1}}|u(t)|^\frac{k+m+1}{m+1}|u_t(t)|^mdx\\ &\leq C_0 (\int_{\Omega}|u(t)|^k|u_t(t)|^{m+1}dx)^\frac{m}{m+1}(\int_{\Omega}|u(t)|^{k+m+1}dx)^\frac{1}{m+1}\\ &\leq C_0B_0(H'(t))^\frac{m}{m+1}\|u(t)\|^\frac{k+m+1}{m+1}_{p+1}\\ &\leq C_0B_0(\frac{1}{\delta}H'(t)+\delta^m\|u(t)\|^{k+m+1}_{p+1}), \end{aligned}\label{3.6} where $\delta$ is a constant to be chosen later, $B_0$ is the embedding constants from $L^{k+m+1}(\Omega)$ to $L^{p+1}(\Omega)$(since $k+m0$ is a constant which does not depended on $\epsilon$. In particular, \eqref{3.12} shows that $y(t)$ is increasing on $(0,T)$, with \begin{equation*} y(t)=H^{1-\alpha}(t)+\epsilon F'(t)\geq H^{1-\alpha}(0)+\epsilon F'(0). \end{equation*} We further choose $\epsilon$ sufficiently small such that $y(0)>0$, so $y(t)\geq y(0)>0$ for $t\geq 0$. Now, let $r=\frac{1}{1-\alpha}$. Since $0<\alpha<\frac{1}{2}$, it is evident that $r>1$. Using Young's inequality again \begin{aligned} y^r(t)&\leq 2^{r-1}(H(t)+\epsilon \|u(t)\|^r\|u_t(t)\|^r)\\ &\leq C_4(H(t)+\|u_t(t)\|^2+\|u(t)\|^{\frac{1}{\frac{1}{2}-\alpha}}).\label{3.13} \end{aligned} By the choice of $\alpha$, we have $\frac{1}{2}-\alpha>\frac{1}{p+1}$. Now apply the inequality \begin{equation*} x^\sigma \leq (1+\frac{1}{a})(a+x),\quad x\geq 0,\quad 0\leq \sigma \leq 1,\quad a>0, \end{equation*} and take $x=\|u(t)\|^{p+1},\sigma=\frac{1}{(\frac{1}{2}-\alpha)(p+1)}<1,a=H(0)$, and $d=1+\frac{1}{H(0)}$, we obtain $$\|u(t)\|^\frac{1}{\frac{1}{2}-\alpha}\leq d(H(0)+\|u(t)\|^{p+1})\leq C_5(H(t)+\|u(t)\|_{p+1}^{p+1}). \label{3.14}$$ Hence, from \eqref{3.13} and \eqref{3.14} there results $$y^r(t)\le C(H(t)+\|u_t(t)\|^2+\|u(t)\|_{p+1}^{p+1}).\label{3.15}$$ Thus, \eqref{3.12} and \eqref{3.15} show that $y'(t)\geq C_6y^r(t),\quad t\in [0,T].$ Finally, from this inequality and $r=\frac{1}{1-\alpha}>1$, we see that $y(t)=H^{1-\alpha}(t)+\epsilon F'(t)$ blow up in finite time. This completes the proof. \end{proof} \section{Asymptotic stability of the solutions} To obtain the asymptotic stability of the solution, we start with a series of lemmas. The assumption of Theorem \ref{thm2.2} will be valid throughout this section. \begin{lemma} \label{lem4.1} If $(\|u_0\|_{p+1},E(0))\in \sum_2$, then $$(\|u(t)\|_{p+1},E(t))\in {\sum}_2,\quad t\geq 0. \label{4.1}$$ Moreover $$E(t)\geq\frac{1}{2}\|u_t(t)\|^2+\frac{1}{4}\|\nabla u(t)\|^2, \quad\quad t\geq 0. \label{4.2}$$ \end{lemma} \begin{proof} By \eqref{2.2} and the embedding theorem, for all $t\geq 0$, there holds \begin{aligned} E_2>E(0) & \geq E(t)\geq \frac{1}{2}\|u_t(t)\|^2+\frac{1}{4}\|\nabla u(t)\|^2+\frac{1}{4}B_1^{-2}\|u(t)\|^2_{p+1}-\frac{1}{2}\|u(t)\|_{p+1}^{p+1}\\ &\geq \frac{1}{2}\|u_t(t)\|^2+\frac{1}{4}\|\nabla u(t)\|^2+g(\|u(t)\|_{p+1}), \end{aligned} \label{4.3} where $g(\lambda)=\frac{1}{4}B_1^{-2}\lambda^2-\frac{1}{2}\lambda^{p+1}$, for $\lambda\geq 0$. It is easy to see that $g(\lambda)$ attains its maximum $E_2$ for $\lambda=\lambda_2,\quad g(\lambda)$ is strictly decreasing for $\lambda\geq \lambda_2$ and $g(\lambda)\to {-\infty}$ as $\lambda\to\infty$. By the continuity of $\|u(t)\|_{p+1}$ and $\lambda(0)=\|u_0\|_{p+1}<\lambda_2$, so that $\lambda(t)<\lambda_2$ for all $t\geq 0$. Also, of course, $E(t)0$ is a constant, then there exists $\alpha=\alpha(\beta)>0$ such that $$\|u_t(t)\|^2+\|\nabla u(t)\|^2-\|u(t)\|^{p+1}_{p+1}\geq\alpha, \quad t\geq 0.\label{4.5}$$ \end{lemma} \begin{proof} By the definition of $E(t)$ and $E(t)\geq\beta$, we have $$\|u_t(t)\|^2+\|\nabla u(t)\|^2\geq 2\beta,\quad t\geq 0. \label{4.6}$$ Now suppose that \eqref{4.5} does not hold. From \eqref{4.4}, there is a sequences ${t_n}\subset R^+$ such that \begin{equation*} \|u_t(t_n)\|^2+\|\nabla u(t_n)\|^2-\|u(t_n)\|_{p+1}^{p+1}\geq \|u_t(t_n)\|^2+\frac{1}{2}\|\nabla u(t_n)\|^2\to 0,(n\to\infty). \end{equation*} Then, we get \begin{equation*} \|u_t(t_n)\|^2\to 0, \quad \|\nabla u(t_n)\|^2\to 0,\quad \text{as }n\to\infty. \end{equation*} This is contradiction with \eqref{4.6}. The lemma is proved. \end{proof} \begin{theorem} \label{thm4.5} Assume the conditions of Theorem \ref{thm2.2}, that $(\|u_0\|_{p+1},E(0))\in\sum_2$, and that $u$ is a weak solution to \eqref{1.1}-\eqref{1.3}. Then $$\lim_{t\to \infty}E(t)=0 , \lim_{t\to\infty}\|\nabla u(t)\|=0.\label{4.7}$$ \end{theorem} \begin{proof} Suppose that \eqref{4.7} fails, then there exists $\beta>0$ such that $E(t)\geq\beta$ for all $t\geq 0$ since \eqref{2.2} and $E(t)\geq 0$. Multiplying both sides of \eqref{1.1} by $u$, integrating over $[T,t]\times \Omega$ $(0\frac{1}{2}$, we know $$(u_t(s),u(s))|_{s=T}^{t}\leq C_{11}(\int_T^tds)^\frac{1}{m+1}-\alpha\int_T^tds. \label{4.12}$$ On the other hand, from Holder inequality and Lemma \ref{lem4.3} (2), \begin{equation*} |(u_t(t),u(t))|\leq C_{12}(\|u_t(t)\|^2+\|\nabla u(t)\|^2)<\infty. \end{equation*} In turn, we reach a contradiction with \eqref{4.12} for fixing $T$ when $t\to\infty$. Hence, we derive $\lim_{t\to\infty}E(t)=0$ and $\lim_{t\to\infty}\|\nabla u(t)\|^2=0$ by \eqref{4.2}. This completes the proof. \end{proof} \begin{remark} \label{rmk4.1}\rm The set $\sum_2$ is called stable set. It is smaller than the potential well introduced by Payne and Sattinger\cite{PS1}. Moreover the value $\lambda_2$ in this paper can be chosen larger than now but $\lambda_2<\lambda_1$. \end{remark} \begin{remark} \label{rmk4.2} \rm The method seems general enough to apply to the generate equation \eqref{1.4} with $f(x,u)$ being source term and also let $Q$ and $F$ depending on time but this will be discuss in a future paper. \end{remark} \begin{thebibliography}{00} \bibitem{BLR1} V. Barbu, I. Lasiecka, M. A. Rammaha; Existence and uniqueness of solutions to wave equations with degenerate damping and source terms, {\it Control and Cybernetics}, {\bf 34(3)} (2005), 665-687. \bibitem{BLR2} V. Barbu, I. Lasiecka, M. A. Rammaha; On nonlinear wave equations with degenerate damping and source terms, {\it Trans. Amer. Math. Soc.}, {\bf 357(7)} (2005), 2571-2611. \bibitem{BLR3} V. Barbu, I. Lasiecka, M. A. Rammaha; Blow-up of generalizedsolutions to wave equations with degenerate damping and source terms, {\it Indiana University Mathematics Journal}, preprint, 2006. \bibitem{BV} A. Boccuto, E. Vitillaro; Asymptotic stability for abstract evolution equations and applications to partial differential system, {\it Rend del circolo mathematico di palermo}, {\bf 4(1)} (1998), 25-49. \bibitem{GT} V. Georgiev, G. Todorova; Existence of a solution of the wave equation with nonlinear damping and source terms, {\it J.of Differential Equations}, {\bf 109}(1994), 295-308. \bibitem{Jo} K. Jorgens; Das Anfang swertproblem im Grossen fur eine klasse nichtlinearer wellengleichungen, {\it Math.Z.} {\bf 77}(1961), 295-308. \bibitem{LeSe} H. A. Levine, J. Serrin; Global nonexistence theorems for quasilinear evolution with dissipation, {\it Arch.Rational Mech.Anal}, {\bf 137}(1997), 341-361. \bibitem{LPS} H. A. Levine, P. Pucci, J. Serrin; Some remarks on global nonexistence for nonautonomous abstract evolution equation, {\it Contemporary Mathematics}, {\bf 208}(1997), 253-263. \bibitem{LRS} H. A. Levine, S. R. Park, J. Serrin; Global existence and nonexistence theorem for quasilinear evolution equations of formally parabolic type, {\it J. Differential Equations}, {\bf 142(1)}(1998), 212-229. \bibitem{LS} J. L. Lions, W. A. Strass; Some nonlinear evolution equations, {\it Bull.Soc.Math.France.}, {\bf 93}(1965),43-96. \bibitem{PR} D. R. Pitts, M. A. Rammaha; Global existence and nonexistence theorems for nonlinear wave equations, {\it Indiana University Mathematics Journal}, {\bf 51(6)}(2002), 1479-1509. \bibitem{PS1} L. E. Payne, D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations.{\it Israel Math.J.},{\bf 22}(1975),273-303. \bibitem{PS2} P. Pucci, D. Sattinger; Stability and blow-up for dissipative evolution equations, In: {\em Reaction-diffusion Systems}, Lecture Notes in Pure and Appl. Math. G.Carittv, E.Mitidieri, eds, Dekker, New York, 1997, 299-317. \bibitem{PuSe1} M. P. Pucci, J.Serrin; Asymptotic stability for nonautonomous dissipative wave systems, {\it Communication on pure and Applied Mathematics}, {\bf 49}(1996), 177-216. \bibitem{PuSe2} P. Pucci, J. Serrin; Global nonexistence for abstract evolution equation with positive initial energy, {\it J of Differential Equation}, {\bf 150}(1998), 205-214. \bibitem{PuSe3} P. Pucci, J. Serrin; Local asymptotic stability for dissipative wave systems, {\it Israel J. Math.} {\bf 104}(1998), 29-50. \bibitem{RS} M. A. Rammaha, T. A. Strei; Global existence and nonexistence for nonlinear wave equations with damping and source terms, {\it Trans. Amer. Math. Soc.} {\bf 354(9)}(2002), 3621-3637. \bibitem{Se} I. E. Segal; Nonlinear semigroups, {\it Annals of Math.} {\bf 78}(1963), 339-364. \bibitem{To} G.Todorova; Dynamics of nonlinear wave equations, {\it Math. Meth. Appl. Sci.} {\bf 27}(2004), 1831-1841. \bibitem{Vi1} E. Vitillaro; Global nonexistence theorem for a class of evolution equations with dissipation, {\it Arch Rational Mech. Anal.} {\bf 149}(1999), 155-182. \bibitem{Vi2} E. Vitillaro; Some new result on global nonexistence and blow-up for evolution problem with positive initial energy, {\it Rend. Istit. Mat. Univ Trieste}, {\bf 31(2)}(2000), 245-275. \bibitem{ZC} H. W. Zhang, G. W. Chen; Asymptotic stability for a nonlinear evolution equation, {\it Comment Math. Univ. Carolinae}, {\bf 45(1)}(2004), 101-107. \end{thebibliography} Qingying Hu \newline Department of Mathematics, Henan University of Technology\\ Zhengzhou 450052, China\\ email address: slxhqy@yahoo.com.cn \smallskip Hongwei Zhang \newline Department of Mathematics, Henan University of Technology\\ Zhengzhou 450052, China\\ email addres: wei661@yahoo.com.cn \section*{Editors note: September 10, 2007} A reader informed us that that parts of the introduction were copied from reference [2], without giving the proper credit. Also that the first statement in Lemma 4.3 maybe false; so that Theorem 4.5 has not been proved. The authors agreed to post a new proof, if they succeed in proving the lemma. Errata: Assumption (A1) should include $p>1$. Inequality (2.2) should read $E(t)\leq E(0)$. \end{document}