Resolvent Estimates for Viscoelastic Systems of Extended Maxwell Type and Their Applications

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 5782-5806, October 2024.
Abstract. In the theory of viscoelasticity, an important class of models admits a representation in terms of springs and dashpots. Widely used members of this class are the Maxwell model and its extended version. This paper concerns resolvent estimates for the system of equations for the anisotropic, extended Maxwell model (EMM) and its marginal model; special attention is paid to the introduction of augmented variables. This leads to the augmented system that will also be referred to as the “original” system. A reduced system is then formed which encodes essentially the EMM; it is a closed system with respect to the particle velocity and the difference between the elastic and viscous strains. Based on resolvent estimates, it is shown that the original and reduced systems generate [math]-groups and the reduced system generates a [math]-semigroup of contraction. Naturally, the EMM can be written in an integro-differential form with a relaxation tensor. However, there is a difference between the original and integro-differential systems, in general, with consequences for whether their solutions generate semigroups or not. Finally, an energy estimate is obtained for the reduced system, and it is proven that its solutions decay exponentially as time tends to infinity. The limiting amplitude principle follows readily from these two results.