Energy Decay in the Dynamics of Complex Bodies with Spreading Microstructures Represented by 3D Vectors

We analyze complex bodies with active microstructure described by a vector field with values v is an element of R-3, which complements the displacement u is an element of R-3. We consider linear elastic constitutive structures for macroscopic and microstructural stresses, diffusion of microstructures, and a viscoelastic-substrate-type effect, which has a regularizing influence that we eliminate at a later stage of our analysis. For weak solutions of the balance equations, we prove decay inequalities in W-1,W-2 and also for the p-power of the norms of u and v in L-p spaces under slip-without-friction-type boundary conditions.