Parabolic equations with natural growth approximated by nonlocal equations

Abstract

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is [Formula: see text] where we take, as the most important instance, [Formula: see text] with [Formula: see text] as well as [Formula: see text], [Formula: see text] is a smooth symmetric function with compact support and [Formula: see text] is either a bounded smooth subset of [Formula: see text], with nonlocal Dirichlet boundary condition, or [Formula: see text] itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation.