Communications in Analysis and Geometry

Volume 16 (2008)

Number 4

Blow-up time for a nonlocal diffusion problem with Dirichlet boundary conditions

Pages: 865 – 882

DOI: http://dx.doi.org/10.4310/CAG.2008.v16.n4.a6

Authors

Théodore K. Boni (Institut National Polytechnique Houphouët, Côte d’Ivoire)

Diabate Nabongo (Université d’Abobo-Adjamé, Côte d’Ivoire)

Abstract

This paper concerns the study of the followinginitial-boundary value problem$\[\left\{\begin{array}{@{}ll}\hbox{$u_t=\varepsilon(J*u-u)+f(u)\quad \mbox{in}\quad \Omega\times(0,T),$}\hbox{$u=0\quad \mbox{in}\quad (\mathbb{R}^N-\Omega)\times(0,T),$}\hbox{$u(x,0)=u_{0}(x)>0\quad \mbox{in}\quad \Omega,$}\end{array}\right.\]$where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ withsmooth boundary $\partial\Omega$,$J*u(x,t)=\int_{\mathbb{R}^N}J(x-y)u(y,t)dy,$ J:$\mathbb{R}^N\longrightarrow\mathbb{R}$ is nonnegative,symmetric $(J(z)=J(-z))$, bounded and$\int_{\mathbb{R}^N}J(z)dz=1$, $f(s)$ is positive,increasing, convex function for positive values of $s$ and$\int^{\infty}\frac{ds}{f(s)}\break <\infty$. The initialdata $u_0\in C^{1}(\overline{\Omega})$. We show that if$\varepsilon$ is small enough, the solution of the aboveproblem blows up in a finite time and its blow-up time goesto the one of the solution of the following differentialequation$\[\left\{\begin{array}{@{}ll}\hbox{$\alpha^{\prime}(t)=f(\alpha(t)),\quad t>0$,}\hbox{$\alpha(0)=M$,}\end{array}%\right.\]$as $\varepsilon$ goes to zero, where$M=\sup_{x\in\Omega}\,u_{0}(x)$.

Finally, we give some numerical results to illustrate ouranalysis.

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