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A third order in time nonlinear equation is considered. This particular model is motivated by High Frequency Ultra Sound (HFU) technology which accounts for thermal and molecular relaxation. The resulting equations give rise to a quasilinear-like evolution with a potentially degenerate damping [23]. The purpose of this paper is twofold: (1) to provide a brief review of recent results in the area of long time behavior of solutions to of MGT equation, (2) to provide recent results pertaining to decay of energy associated with the model accounting for molecular relaxation which is locally distributed. [ABSTRACT FROM AUTHOR]