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Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound

Kaltenbacher B., Lasiecka I., Marchand R.

Control and Cybernetics

2011

Vol. 40, no 4

971-988

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 and instability to exponential stability (in time). It will be shown that by neglecting diffusivity of the sound coefficient there arises a lack of existence of a semigroup associated with the linear dynamics. More specifically, the corresponding linear dynamics consists of three diffusions: two backward and one forward. When diffusivity of the sound is positive, the linear dynamics is described by a strongly continuous semigroup which is exponentially stable when the ratio of sound speed×relaxation parameter/ sound diffusivity is sufficiently small, and unstable in the complementary regime. The theoretical estimates proved in the paper are confirmed by numerical validation.