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

Abstract

We study a thermal model associated with a heat-conducting material based on a three-phase-lag constitutive equation for the heat flux, a model that leads to a Moore–Gibson–Thompson type equation for the thermal displacement. We are researching the compatibility of the three-phase-lag constitutive equation in concern with the second law of thermodynamics, thus discovering restrictions to be imposed on the involved thermal coefficients. On this basis, we manage to obtain the well-posedness problem of the model as the uniqueness of the solutions and their continuous dependence on the given data. Finally, we show that such a model not only allows the propagation of damped in time waves but also exponentially decaying in time thermal standing mode waves. We also show that if the thermodynamic restrictions are not fulfilled, then we can be led to instability. Through the present treatment of the thermal model in question, we obtain important information on the associated Moore–Gibson–Thompson type equation for the thermal displacement.

Spatial Behavior of Solutions in Linear Thermoelasticity with Voids and Three Delay Times

“…the thermodynamic (second law) compatibility constraints of which have been recently analyzed in [4].…”

“…Proceeding in chronological order, we would like to remember, for instance, the work of Ciarletta and Ieşan [16], together with the references by Ignaczak [17] and Ignaczak and coworkers [18] cited therein; also, in [18], we underline the existence of a domain of influence, which is linked to the presence of a relaxation time; or even the works by Hetnarski and Ignaczak [19], Quintanilla and Racke [20] (In which one can read: ... the spatial behavior of solutions is analyzed in a semi-infinite cylinder (framework also applicable to our analysis, see note at the end of Section 4) and a result on the domain of influence is obtained); and, more recently, Ostoja-Starzewski and Quintanilla [21], where the spatial behavior of solutions is investigated, highlighting an interesting parallel with the Moore-Gibson-Thompson (MGT) equation (see also [4,22]). More generally, as Fernández and Quintanilla state in [23]: Mathematical studies about the spatial behavior have been proposed for elliptic, hyperbolic and parabolic equations [...].…”