Nonlocal evolution equations in perforated domains

In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form u t ( t , x ) = ∫ R N ∖ A ϵJ ( x − y ) ( u ( t , y ) − u ( t , x ) ) d y + f ( t , x ) with x in a perturbed domain Ω ϵ ⊂ Ω . We think about Ω ε as a fixed set Ω from where we have removed the subset A ε that we call the holes. Moreover, we take J as a nonsingular kernel. Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω ε has a weak limit, χ ϵ ⇀ X weakly ∗ in L ∞ (Ω) as ε →0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation.