Equipartition of energy for nonautonomous wave equations


Consider wave equations of the form u"(t) + A2u(t) = 0 with A an injective selfadjoint operator on a complex Hilbert space H. The kinetic, potential, and total energies of a solution u are K(t) = ∥u'(t)∥2; P(t) = ∥Au(t)∥2; E(t) = K(t) + P(t): Finite energy solutions are those mild solutions for which E(t) is finite. For such solutions E(t) = E(0), that is, energy is conserved, and asymptotic equipartition of energy lim/t→±∞ K(t) = lim/t→±∞P(t) = E(0)/2 holds for all finite energy mild solutions iff eitA → 0 in the weak operator topology. In this paper we present the first extension of this result to the case where A is time dependent.

Publication Title

Discrete and Continuous Dynamical Systems - Series S