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

1. The main aim of this report is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the nearly isotropic case. In order to state the problem, we recall that the system of linear elasticity is of form $\frac{\partial^{2}u_{i}}{\partial t^{2}}=.\sum_{\wedge-}^{3}$. $c_{ijpq^{\frac{\partial^{2}u_{p}}{\partial x_{j}\partial x_{q}}}}$ , $i=1,$ 2, 3 (1) For crystals, the $c_{ejpq}$ will be real constants (called " stiffness " constants) and depend on the crystal under consideration. At a first glance the system thus apparently depends on the 81 $(=3^{4})$ constants $c_{ijpq}$ , but since the $c_{ijpq}$ have to satisfy the conditions $c_{ijpq}=c_{jipq}=c_{ijqp}=c_{pqij}$ , $\forall i$ , $\forall j$ , $\forall p$ , $\forall q$ , the number of " essential " constants is at most 21. Additional restrictions will come from the fact that the system has to be hyperbolic (the condition is that the form $\sigma=(\sigma_{ij})arrow$ $\sum 3$ the number of essential constants will decrease further when the crystal under consideration has additional symmetries. Thus, when the crystal is cubic, as we will suppose in this report, then the number of essential constants is 3. (Cf. [13].) With the system (1) we now associate the initial conditions: