Extinction time for some nonlinear heat equations

Abstract

This paper concerns the study of the extinction time of the solution of the following initial-boundary value problem [left{% begin{array}{ll} hbox{$u_t=varepsilon Lu(x,t)-f(u)quad mbox{in}quad Omegatimesmathbb{R}_{+}$,} \ hbox{$u(x,t)=0quad mbox{on}quadpartialOmegatimesmathbb{R}_{+}$,} \ hbox{$u(x,0)=u_{0}(x)>0quad mbox{in}quad Omega$,} \ end{array}%right. ] where $Omega$ is a bounded domain in $mathbb{R}^{N}$ with smooth boundary $partialOmega$, $varepsilon$ 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^{\u27}(t)=-f(alpha(t)),quad t>0,quad alpha(0)=M,] as $varepsilon$ goes to zero, where $M=sup_{xin Omega}u_{0}(x)$. We also extend the above result to other classes of nonlinear parabolic equations. Finally, we give some numerical results to illustrate our analysis