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In this paper we consider the non local evolution equation $$ \frac{\partialu(x,t)}{\partial t} + u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)dy+h(x). %\,\,\, h \geq 0. $$ We show that this equation defines a continuous flowin both the space $C_{b}(\mathbb{R}^{N})$ of bounded continuous functions andthe space $C_{\rho}(\mathbb{R}^{N})$ of continuous functions $u$ such that $u\cdot \rho$ is bounded, where $\rho $ is a convenient "weight function"'. Weshow 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. Wethen exhibit a continuous Lyapunov function which is well defined in the wholephase space and continuous in the $C_{\rho}(\mathbb{R}^{N})$ topology, allowingthe characterization of the attractor as the unstable set of the equilibriumpoint set. We also illustrate our result with a concrete example.

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