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

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In this paper we consider the nonlocal 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.