Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition

Abstract

In this paper we consider the following nonlocal autonomous evolution equation in a bounded domain $\hat{I}\text{c◯}$ in ${\mathbb{R}}^N$ \[ \partial_t u(x,t) =- h(x)u(x,t) + g \Big(\int_Ω J(x,y)u(y,t)dy \Big) +f(x,u(x,t)) \] where $h\in W^{1,\infty }(\hat{I}\text{c◯})$, $g:\mathbb{R}\to \mathbb{R}$ and $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ are continuously differentiable function, and $J$ is a symmetric kernel; that is, $J(x,y)=J(y,x)$ for any $x,y\in {\mathbb{R}}^N$. Under additional suitable assumptions on $f$ and $g$, we study the asymptotic dynamics of the initial value problem associated to this equation in a suitable phase spaces. More precisely, we prove the existence, and upper semicontinuity of compact global attractors with respect to kernel $J$.