A gradient flow generated by a nonlocal model of a neural field in an unbounded domain

Abstract

In this paper we consider the non local evolution equation $$\frac{\partial u(x,t)}{\partial t}+u(x,t)={\int }_{{\mathbb{R}}^N}J(x-y)f(u(y,t))\rho (y)dy+h(x).$$ We show that this equation defines a continuous flow in both the space $C_b({\mathbb{R}}^N)$ of bounded continuous functions and the space $C_{\rho }({\mathbb{R}}^N)$ of continuous functions $u$ such that $u\cdot \rho$ is bounded, where $\rho$ is a convenient "weight function"'. We show the existence of an absorbing ball for the flow in $C_b({\mathbb{R}}^N)$ and the existence of a global compact attractor for the flow in $C_{\rho }({\mathbb{R}}^N)$, under additional conditions on the nonlinearity. We then exhibit a continuous Lyapunov function which is well defined in the whole phase space and continuous in the $C_{\rho }({\mathbb{R}}^N)$ topology, allowing the characterization of the attractor as the unstable set of the equilibrium point set. We also illustrate our result with a concrete example.

16 pages