On the stability problem for the exterior linear viscoelasticity (Q1115670)

The stability problem of the solutions of the equations of anisotropic linear viscoelasticity is discussed. After assuming that there exist classical solutions of the viscoelastic equilibrium problem for materials of relaxation and creep type in the exterior domains, it is proved that they are stable with respect to the measures: \[ \mu_ 1(\nu)=\int_{D}(\rho (x)\nu^ 2(x,t)+C_{ijkl}(\nu,t)\nu_{i,j}(x,t)\nu_{k,l}(x,t))dx \] \[ \mu_ 2(\nu)=\int^{t}_{0}d\tau \int_{D}\nu_{i,j}(x_ 1,\tau)\nu_{i,j}(x,\tau)dx,\quad \mu_ 2(\nu_ 0)=\mu_ 1(\nu_ 0), \] where, \(\nu\), C, \(\rho\) and D are, respectively, displacement, elasticity tensor, material density and domain of motion. To obtain the weighted energy equality, Lebesgue's dominated convergence theorem is applied. Using that equality two stability theorems for relaxation type materials and one theorem for creep type materials, are proved.