BLOW-UP TIME OF SOME NONLINEAR WAVE EQUATIONS
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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.
MSC
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