N’gohisse, Firmin K.; Boni, ThĂ©odore K. Quenching time of some nonlinear wave equations. (English) Zbl 1212.35016 Arch. Math., Brno 45, No. 2, 115-124 (2009). 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. Cited in 4 Documents 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 Keywords:nonlinear wave equation; quenching; convergence; numerical quenching time × Cite Format Result Cite Review PDF Full Text: EuDML EMIS