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

TL;DR

This paper investigates the asymptotic behavior of a class of nonlocal evolution equations with Neumann boundary conditions, proving the existence and upper semicontinuity of global attractors as the kernel varies.

Contribution

It establishes the existence and upper semicontinuity of global attractors for nonlocal evolution equations with respect to the kernel function.

Findings

Existence of global attractors for the nonlocal evolution equation.

Upper semicontinuity of attractors with respect to kernel J.

Conditions under which the asymptotic dynamics are well-behaved.

Abstract

In this paper we consider the following nonlocal autonomous evolution equation in a bounded domain $\Omega$ in $\mathbb{R}^N$ \[ \partial_t u(x,t) =- h(x)u(x,t) + g \Big(\int_{\Omega} J(x,y)u(y,t)dy \Big) +f(x,u(x,t)) \] where $h\in W^{1,\infty}(\Omega)$, $g: \mathbb{R} \to \mathbb{R}$ 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 $J$.