Asymptotic behavior for a non-autonomous model of neural fields with variable external stimuli

Abstract

In this work we consider the class of nonlocal non-autonomous evolution problems in a bounded smooth domain $\Omega$ in $\mathbb{R}^{N}$ $$\displaylines{ \partial_t u(t,x) =- a(t)u(t,x) + b(t) \int_{\mathbb{R}^N} J(x,y)f(t,u(t,y))\,dy -h +S(t,x),\quad t\geq\tau \cr u(\tau,x)=u_\tau(x), }$$ with u(t,x)= 0 for $t\geq\tau$ and $x \in\mathbb{R}^N\backslash\Omega$. Under appropriate assumptions we study the asymptotic behavior of the evolution process, generated by this problem in a suitable Banach space. We prove results on existence, uniqueness and smoothness of the solutions and on the existence of pullback attractor for the evolution process. We also prove a continuous dependence of the evolution process with respect to the external stimuli function present in the model. Furthermore, using the continuous dependence of the evolution process, we prove the upper semicontinuity of pullback attractors with respect to the external stimuli function. We finish this article with a small discussion about the model and about a biological interpretation of the result on the continuous dependence of neuronal activity with respect to the external stimuli function.

Keywords

• nonlocal evolution equation
• neural fields
• pullback attractors
• continuous dependence