Title: 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 $$ u_t =\displaystyle \int_{\mathbb{R}^N} J(x-y)
\big( u(y,t) -u(x,t) \big) \mathcal G\big( u(y,t) -u(x,t) \big) dy \qquad
\mbox{ in } \, \Omega \times (0,T)\,, $$ being $ u (x,t)=0 \mbox{ in }
(\mathbb{R}^N\setminus \Omega )\times (0,T)\,$ and $ u(x,0)=u_0 (x) \mbox{ in }
\Omega$. We take, as the most important instance, $\mathcal G (s) \sim 1+
\frac{\mu}{2} \frac{s}{1+\mu^2 s^2 }$ with $\mu\in \mathbb{R}$ 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 $\mathbb{R}^N$, with nonlocal
Dirichlet boundary condition, or $\mathbb{R}^N$ 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.