Generalised Energy Conservation Law for 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 (1.1) utt − a (t)∆u = 0, u(0, ·) = u1 ∈ H (R), Dtu(0, ·) = u2 ∈ L (R) for a wave equation with variable propagation speed. As usual we denote Dt = −i∂t, ∆ = ∑ j ∂ 2 xj the Laplacian on R n 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 [1], [2]), but satisfying a suitable stabilisation condition in the spirit of Hirosawa [3], [4]. For this we assume that the coefficient a(t) can be written as product (1.2) a(t) = λ(t)ω(t) of a shape function λ(t) (being 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 [3] including the shape function λ(t). Roughly speaking, this means that the adapted hyperbolic energy of the solution u(t, x) of (1.1), (1.3) Eλ(t;u) = 1 2 ∫ Rn ( λ(t)|∇u(t, x)| + |ut(t, x)| 2 ) dx satisfies a two-sided energy inequality of the form (1.4) C1 ≤ 1 λ(t) Eλ(t;u) ≤ C2 with constants C1 and C2 depending on the data. The upper bound can be given in terms of the norms of u1 ∈ H 1(Rn) and u2 ∈ L 2(Rn), it is not possible to replace H1(Rn) by the corresponding homogeneous space Ḣ1(Rn) (as in the case of [3]). Supported by KAKENHI (19740072) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Supported by EPSRC grant EP/E062873/1 from the Engineering and Physics Research Council, UK.. 1 2 FUMIHIKO HIROSAWA AND JENS WIRTH The behaviour of the energy is only of interest as t → ∞ (or in the neighbourhood of zeros of λ(t), which is not within the scope of this note). Therefore it is reasonable to restrict considerations to monotonous λ(t) with λ(0) > 0. Basic assumptions of our approach are that a(t) ∈ C(R+), m ≥ 2, together with (A1): λ(t) > 0, λ′(t) > 0 together with the estimates (1.5) λ(t) ≈ λ(t) ( λ(t) Λ(t) ) , |λ(t)| . λ(t) ( λ(t) Λ(t) )2 , where Λ(t) = 1 + ∫ t 0 λ(s)ds denotes a primitive of λ(t); (A2): 0 < c1 ≤ ω(t) ≤ c2; (A3): ω(t) λ-stabilises towards 1, i.e. we assume that