In this paper we consider the non local evolution equation $$ \frac{ \partial u(x,t)}{ \partial t} + u(x,t)= \int_{ \mathbb{ R}^{ N}}J(x-y)f(u(y,t))\rho(y)dy+ h(x). %\,\,\, h \geq 0. $$ 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.
Categories: math.DS