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.