# BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS

Volume 1, Issue 2, pp 91-101
• 1309 Views ### Authors

THEODORE K. BONI - Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, (Cote d'Ivoire).. DIABATE NABONGO - Universite d'Abobo-Adjame, UFR-SFA, Departement de Mathematiques et Informatiques, 16 BP 372 Abidjan 16, (Cote d'Ivoire). ROGER B. SERY - Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, (Cote d'Ivoire)..

### Abstract

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.

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##### ISRP Style

THEODORE K. BONI, DIABATE NABONGO, ROGER B. SERY, BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS, Journal of Nonlinear Sciences and Applications, 1 (2008), no. 2, 91-101

##### AMA Style

BONI THEODORE K., NABONGO DIABATE, SERY ROGER B., BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS. J. Nonlinear Sci. Appl. (2008); 1(2):91-101

##### Chicago/Turabian Style

BONI, THEODORE K., NABONGO, DIABATE, SERY, ROGER B.. "BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS." Journal of Nonlinear Sciences and Applications, 1, no. 2 (2008): 91-101

### Keywords

• Nonlinear wave equation
• blow-up
• convergence
• numerical blow-up time.

•  35B40
•  35B50
•  35K60

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