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In this paper we study several aspects related with solutions of nonlocalproblems 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 nonlocalDirichlet boundary condition, or $\mathbb{R}^N$ itself.  The results deal with existence, uniqueness, comparison principle andasymptotic behavior. Moreover we prove that if the kernel rescales in asuitable way, the unique solution of the above problem converges to a solutionof the deterministic Kardar-Parisi-Zhang equation.

Parabolic equations with natural growth approximated by nonlocal equations