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

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