Abstract
As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u tt+u t=Δu+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 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 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.
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Gallay, T., Slijepčević, S. Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains. Journal of Dynamics and Differential Equations 13, 757–789 (2001). https://doi.org/10.1023/A:1016624010828
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DOI: https://doi.org/10.1023/A:1016624010828