Optimal scalar products in the Moore-Gibson-Thompson equation

Abstract

We study the third order in time linear dissipative wave equation known as the Moore-Gibson-Thompson equation, that appears as the linearization of a the Jordan-Moore-Gibson-Thompson equation, an important model in nonlinear acoustics. The same equation also arises in viscoelasticity theory, as a model which is considered more realistic than the usual Kelvin-Voigt one for the linear deformations of a viscoelastic solid. In this context, it is known as the Standard Linear Viscoelastic model. We complete the description in [13] of the spectrum of the generator of the corresponding group of operators and show that, apart from some exceptional values of the parameters, this generator can be made to be a normal operator with a new scalar product, with a complete set of orthogonal eigenfunctions. Using this property we also obtain optimal exponential decay estimates for the solutions as t → ∞, whether the operator is normal or not.

Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

“…In particular, the associated semigroup on the natural weak energy space is exponentially stable if and only if we are in the subcritical regime, corresponding to χ > 0 (see e.g. [12,23,31,39,40]…”

“…where A is a strictly positive operator on some Hilbert space H and α, β, γ > 0 are given parameters, has deserved much attention in recent years, with several papers appeared in the literature on the argument (see [3,11,12,13,23,25,31,39,40], among others). The model has been originally introduced in connection with fluids mechanics [50].…”

Moore–Gibson–Thompson thermoelasticity

“…The Moore-Gibson-Thompson equation (MGT, for short) has deserved a big interest in the recent years and many papers has been directed to study and understand it (see among others [4,12,14,21,28,29]). It arises from the uids mechanics [38].…”

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

“…Over the last years, the Moore -Gibson -Thompson (MGT) equation (cf. [30,45]), a linearization of a model for the wave propagation in viscous thermally relaxing fluids, has been studied by several authors (see, for example, [13,20,19,28,18,27,7,11,26,12,25,37,3,10,38,5,6,4,8]).…”