The Galerkin Method for Fourth-Order Equation of the Moore–Gibson–Thompson Type with Integral Condition

Research on the nonlinear propagation of sound in a situation of high amplitude waves has shown literature on physically well-founded partial differential models (see, e.g., [1–23]). This still very active field of research is carried by a wide range of applications such as the medical and industrial use of high-intensity ultrasound in lithotripsy, thermotherapy, ultrasound cleaning, and sonochemistry. The classical models of nonlinear acoustics are Kuznetsov’s equation, the Westervelt equation, and the KZK (Kokhlov-Zabolotskaya-Kuznetsov) equation. For a mathematical existence and uniqueness analysis of several types of initial boundary value problems for these nonlinear second order in time PDEs, we refer to [24–44]. Focusing on the study of the propagation of acoustic waves, it should be noted that the MGT equation is one of the equations of nonlinear acoustics describing acoustic wave propagation in gases and liquids. The behavior of acoustic waves depends strongly on the medium property related to dispersion, dissipation, and nonlinear effects. It arises from modeling high-frequency ultrasound (HFU) waves (see [10, 12, 34]). The derivation of the equation, based on continuum and fluid mechanics, takes into account viscosity and heat conductivity as well as effect of the radiation of heat on the propagation of sound. The original derivation dates back to [44]. This model is realized through the third-order hyperbolic equation: