Ricerc@Sapienza

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.