Slow Decay in Linear Thermo-elasticity

Energy estimates show that linearized thermo-elasticity deenes a contraction semi-group on a Hilbert space. We show that under a geometric condition this contraction is not strict, or, more precisely, the norm of the semi group is 1 for all t 0. Convex domains always satisfy the geometric condition. 2. introduction Let R n be a bounded domain with smooth boundary ?. We consider the problem @ 2 tt u ? u ? (+)r(ru) + r = 0 in R + c p @ t ? + @ t ru = 0 in R + (2.1) where u is a time dependent deformation vector eld, is the normalized temperature, , , and c p are positive constants, 2 + > 0 (the density, and are Lame's constants and c p is the speciic heat), is the coupling constant between pressure waves and heat, r(ru) = grad divu, and u denotes the component wise Laplacian. This system is complemented by initial conditions and the boundary condition