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

u_tt+2F(S)u_t+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.