Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation

Abstract

[EN] We study a third-order partial differential equation in the form $\tau u_{ttt} +\alpha u_{tt} -c^2 u_{xx} -b u_{xxt} =0, (1)$$ that corresponds to the one-dimensional version of the Moore-Gibson-Thompson equation arising in high-intensity ultrasound and linear vibrations of elastic structures. In contrast with the current literature on the subject, we show that when the critical parameter $\gamma:=\alpha-\frac{\tauc^2}{b}$ is negative, the equation (1) admits an uniformly continuous, chaotic and topologically mixing semigroup on Banach spaces of Herzog s type.