Time decay for several porous thermoviscoelastic systems of Moore–Gibson–Thompson type

Abstract

In this paper, we consider several problems arising in the theory of thermoelastic bodies with voids. Four particular cases are considered depending on the choice of the constitutive tensors, assuming different dissipation mechanisms determined by Moore–Gibson–Thompson-type viscosity. For all of them, the existence and uniqueness of solutions are shown by using semigroup arguments. The energy decay of the solutions is also analyzed for each case.

On the decay of the energy for radial solutions in Moore–Gibson–Thompson thermoelasticity

“…It is worth noting that we can also obtain the MGT system of thermoelasticity as a particular case of the theory proposed by Gurtin [15], as can be seen in [16, 17]. It is also worth remarking that this thermoelastic theory has received much attention in the last 2 years (see [9, 16–27], among others).…”

On a three-phase-lag heat conduction model for rigid conductor

“…It was observed by Quintanilla [7] that the Moore-Gibson-Thompson equation appears in the generalized thermoelastic theories based upon the introduction of the thermal displacement (see Green-Naghdi's model [8][9][10]) and using the Maxwell-Cattaneo constitutive equation for the heat flux vector. Besides, there is a recent, rich, and intense research activity of the third-order in time differential equations, either separately or coupled with the elastic deformations of medium, leading to valuable results concerning the well-posedness: Quintanilla [7], Pellicer and Quintanilla [11], Conti et al [12], Ostoja-Starzewski and Quintanilla [13], Bazarra et al [14], Fernandez and Quintanilla [15,16], and Fernandez et al [17]. All these studies suffer from an inconvenience, namely, that they prescribe initial and boundary data in terms of temperature (or thermal displacement), instead of including mixed data for temperature (or thermal displacement) and heat flux vector.…”

A Moore‐Gibson‐Thompson heat conduction equation for non centrosymmetric rigid solids

“…In this work, we want to focus on the theory of heat conduction called Moore-Gibson-Thompson. It is worth recalling that this theory has received much attention in the last four years [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] since many authors have investigated the qualitative and quantitative properties of the solutions to this equation. If we consider the type III heat conduction theory proposed by Green-Naghdi [6], we can see that it is based on the constitutive equation in the case of centrosymmetric materials:…”

“…In this work, we want to focus on the theory of heat conduction called Moore‐Gibson‐Thompson. It is worth recalling that this theory has received much attention in the last four years [10–24] since many authors have investigated the qualitative and quantitative properties of the solutions to this equation. If we consider the type III heat conduction theory proposed by Green‐Naghdi [6], we can see that it is based on the constitutive equation in the case of centrosymmetric materials: $$$$\begin{equation*} q_i=k_{ij} \alpha _{,j}+ \kappa _{ij}^* \theta _{,j}, \end{equation*}$$$$where $$q_i$$ is the heat flux vector, θ is the temperature and α is the thermal displacement.…”