Large Time Behavior of a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data
Abstract
We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(xy)(u(y,t)u(x,t))\,dyu^p$, $u(x,0)=u_0(x)\in L^\infty$, where $x^{\alpha}u_0(x)\to A>0$ as $x\to\infty$. One of our main goals is the study of the critical case $p=1+2/\alpha$ for $0<\alpha<N$, left open in previous articles, for which we prove that $t^{\alpha/2}u(x,t)U(x,t)\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}x^{\alpha}$. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data $u_0$ in the supercritical case and also in the critical case ($p=1+2/N$) for bounded and integrable $u_0$.
 Publication:

arXiv eprints
 Pub Date:
 April 2010
 DOI:
 10.48550/arXiv.1004.0717
 arXiv:
 arXiv:1004.0717
 Bibcode:
 2010arXiv1004.0717T
 Keywords:

 Mathematics  Analysis of PDEs