#### Mathematical Communications, Vol. 13 No. 2, 2008.

Izvorni znanstveni članak

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:

str. 241-251

preuzimanja: 588

###### Sažetak

This paper concerns the study of the extinction time of the solution of the following initial-boundary value problem
$\left\{% \begin{array}{ll} \hbox{u_t=\varepsilon Lu(x,t)-f(u)\quad \mbox{in}\quad \Omega\times\mathbb{R}_{+},} \\ \hbox{u(x,t)=0\quad \mbox{on}\quad\partial\Omega\times\mathbb{R}_{+},} \\ \hbox{u(x,0)=u_{0}(x)>0\quad \mbox{in}\quad \Omega,} \\ \end{array}%\right.$
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.

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