ABSTRACT
In this chapter we shall describe in detail the behavior of solutions to the linearized equations in two or three space dimensions as time tends to infin ity. The picture is not as complete as in the one-dimensional case which was discussed in the previous chapter; in one space dimension the asymptotic behavior is predominated by the dissipation through heat conduction. In all situations (different types of domains) the decay of solutions is exactly the same as for solutions to pure heat equations, both for the temperature difference and the displacement. Now we shall see that in more than one space dimension the hyperbolic part dominates, and the generic situation will be that there is no decay of the associated semigroup but oscillations appear that remain until infinity. Only in special situations like the radially symmetric case in bounded domains, having rotation of the displacement vector identical to zero, can a decay result and then even an exponential decay be proved.
