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Quenching time of some nonlinear wave equations. (English) Zbl 1212.35016
Summary: In this paper we consider the following initial-boundary value problem \( u_{tt}(x,t)=\varepsilon Lu(x,t)+f\big (u(x,t)\big )\) in \(\Omega \times (0,T)\), \( u(x,t)=0 \) on \(\partial \Omega \times (0,T)\) and zero initial data, 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 \(s\in (-\infty ,b)\), \(\lim _{s\to b}f(s)=\infty \) and \(\int _0^b1/f(s)\, ds <\infty \) with \(b=\text{const}>0\). 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 \(\alpha ^{\prime \prime }(t)=f(\alpha (t)),\, t>0\) with zero initial data 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.

35B40 Asymptotic behavior of solutions to PDEs
35B50 Maximum principles in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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