Anisotropic thermo-elasticity in 2D. Part II: Applications

This note deals with concrete applications of the general treatment of anisotropic thermo-elasticity developed in the first part [M. Reissig, J. Wirth, Anisotropic thermo-elasticity in 2D A unified treatment, Asympt. Anal. ?? (????) ??–??]. We give dispersive decay rates for solutions to the type-1 system of thermo-elasticity for certain types of anisotropic media. 1. Elastic operators and thermo-elastic systems In the first part, [2] we treated the type-1 system of thermo-elasticity for an anisotropic medium in two space dimensions. This system reads as Utt +A(D)U + γ∇θ = 0 (1.1a) θt − κ∆θ + γ∇ · Ut = 0 (1.1b) for the elastic displacement U(t, x) ∈ R and the temperature difference to the equilibrium state θ(t, x) ∈ R. Physical properties of the medium are described by the thermal conductivity κ > 0, the thermo-elastic coupling γ > 0 and the elastic operator A(D). Here we assume that its symbol has the structure A(ξ) = D(ξ)SD(ξ), (1.2) where D = ξ1 0 0 ξ2 ξ2 ξ1  (1.3) is a matrix of first order symbols and S = τ1 λ σ1 λ τ2 σ2 σ1 σ2 μ  (1.4) contains the elasticity modules of the medium. Usually one makes the assumption that S is positive definite such that A(D) is a positive and self-adjoint operator. Then the first equation is hyperbolic, while the second one is parabolic. In [2] the thermo-elastic system was treated under the assumptions (A1-2) A(ξ) is positive (self-adjoint, 2-homogeneous and real-analytic in η = ξ/|ξ|); (A3) #spec A(η) = 2 for all η ∈ S; (A4) γ 6= κj(η)− tr A(η) for all hyperbolic directions (with respect to κj(η) ∈ spec A(η)), where a direction η ∈ S is called hyperbolic with respect to κj(η) if the corresponding (normalised) eigenvector rj(η) is perpendicular to the direction η. Under these conditions the system was reformulated as system of first order and Fourier integral representations in terms of Fourier 1 ar X iv :0 70 8. 03 15 v2 [ m at h. A P] 1 5 N ov 2 00 7