Asymptotic equipartition of operator-weighted energies in damped wave equations

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 \u27small\u27. This means that under certain assumptions, the ratio of suitably modified kinetic and potential energies, K̃(t)/P ̃(t), tends to 1 as t→ for all nonzero solutions u(t) of the equation. Here, K̃(t) and P̃(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. © 2013-IOS Press and the authors. All rights reserved