Quenching time of some nonlinear wave equations

Abstract

summary:In this paper, we consider the following initial-boundary value problem {\left\rbrace \begin{array}{ll} u_{tt}(x,t)=\varepsilon Lu(x,t)+f\big (u(x,t)\big )\quad \mbox {in}\quad \Omega \times (0,T)\,,\\ u(x,t)=0 \quad \mbox {on}\quad \partial \Omega \times (0,T)\,, \\ u(x,0)=0 \quad \mbox {in}\quad \Omega \,, \\ u_t(x,0)=0 \quad \mbox {in}\quad \Omega \,, \end{array}\right.} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\varepsilon$ is a positive parameter, $f(s)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, $\lim _{s\rightarrow b}f(s)=\infty$ and $\int _0^b\frac{ds}{f(s)}0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation $${\left\rbrace \begin{array}{ll} \alpha ^{\prime \prime }(t)=f(\alpha (t))\,,&\quad t>0\,, \\ \alpha (0)=0\,,\quad \alpha ^{\prime }(0)=0\,, \end{array}\right.}$$ as $\varepsilon$ goes to zero. We also show that the above result remains valid if the domain $\Omega$ is large enough and its size is taken as parameter. Finally, we give some numerical results to illustrate our analysis