Asymptotic behavior for a nonlocal diffusion equation in domains with holes

C. CORTAZAR, M. ELGUETA, F. QUIROS, N. WOLANSKI

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS

SPRINGER

Lugar: Berlin; Año: 2012 vol. 205 p. 673 - 697

0003-9527

The paper deals with the asymptotic behavior of solutions to anon-local diffusion equation,  $u_t=J*u-u:=Lu$, in an exterior domain,$Omega$, which excludes one or several holes, and with zeroDirichlet data on $mathbb{R}^NsetminusOmega$. When the spacedimension is three or more this behavior is given by a multiple ofthe fundamental solution of the heat equation away from the holes.On the other hand, if the solution is scaled according to its decayfactor, close to the holes it behaves like a function that is$L$-harmonic, $Lu=0$, in the exterior domain and vanishes in its complement.The height of such a function  at infinity is determined through a  matchingprocedure with the multiple of the fundamental solution of the heat equationrepresenting the outer behavior. The inner and the outer behaviorcan be presented in a unified way through a suitable globalapproximation.