The time differential three-phase-lag heat conduction model: Thermodynamic compatibility and continuous dependence

Abstract

This paper deals with the time differential three-phase-lag heat transfer model aiming, at first, to identify the restrictions that make it thermodynamically consistent. The model is thus reformulated by means of the fading memory theory, in which the heat flux vector depends on the history of the thermal displacement gradient: the thermodynamic principles are then applied to obtain suitable restrictions involving the delay times. Consistently with the thermodynamic restrictions just obtained, a first result about the continuous dependence of the solutions with respect to the given initial data and to the supply term is established for the related initial boundary value problems. Subsequently, to provide a more comprehensive review of the problem, a further continuous dependence estimate is proved, this time conveniently relaxing the hypotheses about the above-said thermodynamic restrictions. This last estimate allows the solutions to grow exponentially in time and so to have asymptotic instability.