Parabolic equations with natural growth approximated by nonlocal equations

In this paper we study several aspects related with solutions of nonlocal problems whose prototype is $$ u_t =displaystyle int_mathbbR^N J(x-y) ig( u(y,t) -u(x,t) ig) mathcal Gig( u(y,t) -u(x,t) ig) dy qquad mbox in , Omega imes (0,T),, $$ being $ u (x,t)=0 mbox in (mathbbR^Nsetminus Omega ) imes (0,T),$ and $ u(x,0)=u_0 (x) mbox in Omega$. We take, as the most important instance, $mathcal G (s) sim 1+ racmu2 racs1+mu^2 s^2 $ with $muin mathbbR$ as well as $u_0 in L^1 (Omega)$, $J$ is a smooth symmetric function with compact support and $Omega$ is either a bounded smooth subset of $mathbbR^N$, with nonlocal Dirichlet boundary condition, or $mathbbR^N$ itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover we prove that if the kernel rescales in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar-Parisi-Zhang equation.