99-452 Th. Gallay, S. Slijepcevic

Energy Flow in Extended Gradient Partial Differential Equations (118K, (uuencoded gzipped) Postscript) Nov 30, 99

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Abstract. As an example of an extended, formally gradient dynamical system, we consider a damped hyperbolic equation in R^N with a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space. If N <= 2, 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 deduce that the only uniformly recurrent points are equilibria. Counter-examples are given which show that these results cannot be extended to higher space dimensions.

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