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Extinction time for some nonlinear heat equations

Louis A. Assalé ; Institut National Polytechnique Houphouët-Boigny de Yamoussoukro, Yamoussoukro, Côte d'Ivoire
Théodore K. Boni
Diabate Nabongo

Puni tekst: engleski pdf 145 Kb

str. 241-251

preuzimanja: 588



This paper concerns the study of the extinction time of the solution of the following initial-boundary value problem
\hbox{$u_t=\varepsilon Lu(x,t)-f(u)\quad \mbox{in}\quad \Omega\times\mathbb{R}_{+}$,} \\
\hbox{$u(x,t)=0\quad \mbox{on}\quad\partial\Omega\times\mathbb{R}_{+}$,} \\
\hbox{$u(x,0)=u_{0}(x)>0\quad \mbox{in}\quad \Omega$,} \\
\end{array}%\right. \]
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial\Omega$, $\varepsilon$ is a positive parameter, $f(s)$ is a positive, increasing, concave function for positive values of s, $f(0)=0$, $\int_{0}\frac{ds}{f(s)}<+\infty$, $L$ is an elliptic operator. We show that the solution of the above problem extincts in a finite time and its extinction time goes to that of the solution $\alpha(t)$ of the following differential equation
\[\alpha^{'}(t)=-f(\alpha(t)),\quad t>0,\quad \alpha(0)=M,\] as
$\varepsilon$ goes to zero, where $M=\sup_{x\in \Omega}u_{0}(x)$.
We also extend the above result to other classes of nonlinear
parabolic equations. Finally, we give some numerical results to
illustrate our analysis.

Ključne riječi

extinction, finite difference, nonlinear heat equations, extinction time

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