# Quenching time of some nonlinear wave equations

Archivum Mathematicum (2009)

• Volume: 045, Issue: 2, page 115-124
• ISSN: 0044-8753

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## Abstract

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In this paper, we consider the following initial-boundary value problem $\left\}\begin{array}{c}{u}_{tt}\left(x,t\right)=\epsilon Lu\left(x,t\right)+f\left(u\left(x,t\right)\right)\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{1.0em}{0ex}}\Omega ×\left(0,T\right)\phantom{\rule{0.166667em}{0ex}},\hfill \\ u\left(x,t\right)=0\phantom{\rule{1.0em}{0ex}}\text{on}\phantom{\rule{1.0em}{0ex}}\partial \Omega ×\left(0,T\right)\phantom{\rule{0.166667em}{0ex}},\hfill \\ u\left(x,0\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{1.0em}{0ex}}\Omega \phantom{\rule{0.166667em}{0ex}},\hfill \\ {u}_{t}\left(x,0\right)=0\phantom{\rule{1.0em}{0ex}}\text{in}\phantom{\rule{1.0em}{0ex}}\Omega \phantom{\rule{0.166667em}{0ex}},\hfill \end{array}\right\$ where $\Omega$ is a bounded domain in ${ℝ}^{N}$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\epsilon$ is a positive parameter, $f\left(s\right)$ is a positive, increasing, convex function for $s\in \left(-\infty ,b\right)$, ${lim}_{s\to b}f\left(s\right)=\infty$ and ${\int }_{0}^{b}\frac{ds}{f\left(s\right)}<\infty$ with $b=const>0$. 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\}\begin{array}{cc}{\alpha }^{\text{'}\text{'}}\left(t\right)=f\left(\alpha \left(t\right)\right)\phantom{\rule{0.166667em}{0ex}},\hfill & \phantom{\rule{1.0em}{0ex}}t>0\phantom{\rule{0.166667em}{0ex}},\hfill \\ \alpha \left(0\right)=0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}{\alpha }^{\text{'}}\left(0\right)=0\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}\right\$ as $\epsilon$ 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.

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