EQUIPARTITION OF ENERGY FOR NONAUTONOMOUS WAVE EQUATIONS

Consider wave equations of the form u"(t) + A(2)u(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) = parallel to u'(t)parallel to(2), P(t) = parallel to Au(t)parallel to(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 ->perpendicular to infinity) K(t) = lim(t ->perpendicular to infinity) P(t) = E(0)/2 holds for all finite energy mild solutions iff e(itA) -> 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.