Homogenization for Nonlocal Evolution Problems with Three Different Smooth Kernels

In this paper we consider the homogenization of evolution problem associated with a jump process that involves three different smooth kernels govern jumps to/from parts domain. We assume spacial domain is divided into sequence two subdomains $$A_n \cup B_n$$ and have kernels, one controls from $$A_n$$ to , second $$B_n$$ third governs interactions between . Assuming $$\chi _{A_n} (x) \rightarrow X(x)$$ weakly in $$L^\infty $$ (and then _{B_n} 1-X(x)$$ ) as $$n \infty initial condition given by density $$u_0$$ $$L^2$$ show there an homogenized limit system which function X appear. When delta at point, $$\delta _{{\bar{x}}}$$ (this corresponds starts $${\bar{x}}$$ convergence along subsequences such $${\bar{x}} \in A_{n_j}$$ or B_{n_j}$$ for every $$n_j$$ large enough. also provide probabilistic interpretation equation terms stochastic describes movement particle $$\Omega according underlying converges distribution equation. focus our analysis Neumann type boundary conditions briefly describe end how deal Dirichlet conditions.