  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, abbreviated as the EMM, and its marginal realization which
includes an inertia term; 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
$C_0$-groups and the reduced system generates a $C_0$-semigroup of contraction.
Naturally, the EMM can be written in integrodifferential form leading
explicitly to relaxation and a viscoelastic integro-differential system.
However, there is a difference between the original and integrodifferential
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.
