Global solvability of Moore–Gibson–Thompson equation with memory arising in nonlinear acoustics


A third-order in time nonlinear equation with memory term is considered. This particular model is motivated by high-frequency ultrasound technology which accounts for thermal and molecular relaxation. The resulting equations give rise to a quasilinear-like evolution with a potentially degenerate damping (Kaltenbacher in Evol Eqs Control Theory 4(4):447–491, 2015). Local and global (in time) existence of smooth solutions is studied. The main result of the paper states that with appropriate calibration of the memory kernel, solutions exist globally for sufficiently small and regular initial data. With exponentially decaying relaxation kernel said solutions exhibit exponential decay rates. The proof relies on the “barrier” method applied to a string of higher energy estimates, along with an abstract representation and the theory of viscoleastic evolutions developed in Pruss (Arch Math 92:158–173, 2009, Evolutionary integral equations and applications. Birkhauser, 2012).

Publication Title

Journal of Evolution Equations