Affiliations: University of Memphis, Memphis, TN, USA | Universidad Politécnica de Madrid, Madrid, Spain
Note: [] Corresponding author: Guillermo Reyes, Universidad Politécnica de Madrid, Avd. Ramiro de Maeztu, 7, 28040 Madrid, Spain. E-mail: guillermo.reyes@upm.es
Abstract: We prove an asymptotic energy equipartition result for abstract damped wave equations of the form utt+2F(S)ut+S2u=0, where S is a strictly positive self-adjoint operator and the damping operator F(S) is “small”. This means that under certain assumptions, the ratio of suitably modified kinetic and potential energies, K˜(t)/P˜(t), tends to 1 as t→∞ for all nonzero solutions u(t) of the equation. Here, K˜(t) and P˜(t) are conveniently weighted versions of the usual kinetic and potential energies of the associated undamped equation. Previous results, concerning the undamped case and the scalar-damped one, are particular cases. We propose an extension of the concepts of hyperbolicity and unitarity that allows one to consider the equipartition property in a more general setting. Some examples involving PDEs, as well as pseudo-differential equations, are given.
Keywords: abstract wave equations, energy equipartition
