Decay estimates for the solutions of the system of crystal acoustics for cubic crystals

Abstract

The aim of this paper is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the generic nearly isotropic case. Our main result is that for large time, the solutions to the system decay with a decay rate of order t^{−1/2−1/κ} for some κ∈N which is strictly positive (cf. Remark 1.2). We should observe that in the isotropic case, the optimal decay rate is t⁻¹ and that the decay rate for the wave equation in n space variables is t^−(n−1)/2. On the other hand, the decay rate for solutions of the Maxwell system for optically biaxial crystals is of order t^−1/2. (See O. Liess, Asymptot. Anal. 4 (1991), 61–95.)