# zbMATH — the first resource for mathematics

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.

##### MSC:
 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
Full Text: