BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS

BONI, THEODORE K.¹ ; NABONGO, DIABATE² ; SERY, ROGER B.¹

¹ Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, (Cote d'Ivoire).
² Universite d'Abobo-Adjame, UFR-SFA, Departement de Mathematiques et Informatiques, 16 BP 372 Abidjan 16, (Cote d'Ivoire)

Journal of nonlinear sciences and its applications, Tome 1 (2008) no. 2, p. 91-101.

Résumé

In this paper, we consider the following initial-boundary value problem

$ \begin{cases} u_{tt}(x, t) = \varepsilon Lu(x, t) + b(t)f(u(x, t)) ,\,\,\,\,\, \texttt{in} \qquad\Omega\times (0, T),\\ u(x, t) = 0 ,\,\,\,\,\, \texttt{on} \qquad\partial\Omega\times (0, T),\\ u(x, 0) = 0 ,\,\,\,\,\, \texttt{in}\qquad \Omega,\\ u_t(x, 0) = 0 ,\,\,\,\,\, \texttt{in}\qquad \Omega, \end{cases} $

where $\varepsilon$ is a positive parameter, $b \in C^1(\mathbb{R}_+), b(t) > 0, b' (t)\geq 0, t \in \mathbb{R}_+, f(s) $ is a positive, increasing and convex function for nonnegative values of s. Under some assumptions, we show that, if $\varepsilon$ is small enough, then the solution u of the above problem blows up in a finite time, and its blow-up time tends to that of the solution of the following differential equation

$ \begin{cases} \alpha' (t) = b(t)f(\alpha(t)),\quad t > 0,\\ \alpha(0) = 0, \alpha'(0) = 0. \end{cases} $

Finally, we give some numerical results to illustrate our analysis.

Détail

DOI : 10.22436/jnsa.001.02.05

Classification : 35B40, 35B50, 35K60
Keywords: Nonlinear wave equation, blow-up, convergence, numerical blow-up time.

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