Equipartition of energy for nonautonomous damped wave equations

The kinetic and potential energies for the damped wave equations u'' + 2Bu' + A^2u=0 (DWE) are defined by K(t)=||u'(t)||^2 , P(t)= ||Au(t)||^2, where A,B are suitable commuting seladjoint operators. Asymptotic equipartition of energy means lim_{t o infty} K(t)/P(t)=1 (AEE) for all (finite energy) non-zero solutions of (DWE). The main result of this paper is the proof of a result analogous to (AEE) for a nonautonomous version of (DWE)