Nonlocal and nonlinear evolution equations in perforated domains

Abstract

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form ut(x,t)=∫J(x−y)u(y,t)dy−hε(x)u(x,t)+f(x,u(x,t)) with x in a perturbed domain Ωε⊂Ω which is thought as a fixed set Ω from where we remove a subset Aε called the holes. We choose appropriated families of functions hε∈L∞ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside Ω. Moreover, we take J as a non-singular kernel and f as a nonlocal nonlinearity. Under the assumption that the characteristic functions of Ωε have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.