Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains

As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u_tt+u_t=u+F(x, u) in R^N, where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H¹_ul(R^N)×L²_ul(R^N). If N2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions.