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 $u_t(x,t)=\int J(x-y)u(y,t)dy-h_{\epsilon }(x)u(x,t)+f(x,u(x,t))$ with $x$ in a perturbed domain ${\Omega }^{\epsilon }\subset \Omega$ which is thought as a fixed set $\Omega$ from where we remove a subset $A^{\epsilon }$ called the holes. We choose an appropriated families of functions $h_{\epsilon }\in L^{\infty }$ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside $\Omega$. Moreover, we take $J$ as a non-singular kernel and $f$ as a nonlocal nonlinearity. % Under the assumption that the characteristic functions of ${\Omega }^{\epsilon }$ have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.