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 $$ u_t =\displaystyle \int_{\mathbb{R}^N} J(x-y) \big( u(y,t) -u(x,t) \big) \mathcal G\big( u(y,t) -u(x,t) \big) dy \qquad \mbox{ in } \, Ω\times (0,T)\,, $$ being $ u (x,t)=0 \mbox{ in } (\mathbb{R}^N\setminus Ω)\times (0,T)\,$ and $ u(x,0)=u_0 (x) \mbox{ in } Ω$. 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.