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In the theory of viscoelasticity, an important class of models admits arepresentation in terms of springs and dashpots. Widely used members of thisclass are the Maxwell model and its extended version. This paper concernsresolvent estimates for the system of equations for the anisotropic, extendedMaxwell model, abbreviated as the EMM, and its marginal realization whichincludes an inertia term; special attention is paid to the introduction ofaugmented variables. This leads to the augmented system that will also bereferred to as the "original" system. A reduced system is then formed whichencodes essentially the EMM; it is a closed system with respect to the particlevelocity and the difference between the elastic and viscous strains. Based onresolvent estimates, it is shown that the original and reduced systems generate$C_0$-groups and the reduced system generates a $C_0$-semigroup of contraction.Naturally, the EMM can be written in integrodifferential form leadingexplicitly to relaxation and a viscoelastic integro-differential system.However, there is a difference between the original and integrodifferentialsystems, in general, with consequences for whether their solutions generatesemigroups or not. Finally, an energy estimate is obtained for the reducedsystem, and it is proven that its solutions decay exponentially as time tendsto infinity. The limiting amplitude principle follows readily from these tworesults.

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