Generalised Energy Conservation Law for the Wave Equations with Variable Propagation Speed

We investigate the long time behaviour of the L-energy of solutions to wave equations with variable speed. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property. 1. Model problem We consider the Cauchy problem utt − a (t)∆u = 0, u(0, ·) = u1 ∈ H (R), Dtu(0, ·) = u2 ∈ L (R) (1.1) for a wave equation with variable propagation speed. As usual we denote Dt = −i∂t, ∆ = ∑ j ∂ 2 xj the Laplacian on R and a2(t) is a sufficiently regular non-negative function subject to conditions specified later on. We are interested in the behaviour of the energy as t → ∞ for coefficients bearing very fast oscillations (in the classification of ReissigYagdjian [6], [5]), but satisfying a suitable stabilisation condition in the spirit of Hirosawa [2], [3]. For this we assume that the coefficient a(t) can be written as product a(t) = λ(t)ω(t) (1.2) of a shape function λ(t) (beeing essentially free of oscillations) and a bounded perturbation ω(t) containing a certain amount of oscillations controlled by our main assumptions. Our method leads to an extension of the generalised energy conservation law from [2] including the shape function λ(t). Roughly speaking, this means that the adapted hyperbolic energy of the solution u(t, x) of (1.1),