Exponential decay for low and higher energies in the third order linear Moore-Gibson-Thompson equation with variable viscosity

We consider the Moore-Gibson-Thompson equation which arises, e.g., as a linearization of a model for wave propagation in viscous thermally relaxing fluids. This third order in time equation displays, even in the linear version, a variety of dynamical behaviors for their solutions that depend on the physical parameters in the equation. These range from non-existence [3] and instability to exponential stability (in time) [11]. By neglecting diffusivity of the sound coefficient there is a lack of generation of a semigroup associated with the linear dynamics. When diffusivity of the sound is positive, the linear dynamics is described by a strongly continuous evolution. We shall show that this evolution is exponentially stable provided sufficiently large viscous damping is accounted for in the model. The viscosity considered is time and space dependent which then leads to evolution rather then semigroup generators. Decay rates for both natural and higher level energies are derived.