A Well-Posed Problem for the Dual-Phase-Lag Heat Conduction

The dual-phase lag theory based on the law q(x, t + τ q ) = −K ∇ T(x, t + τ T ) was proposed from an intuitive point of view by Tzou. This equation shows that the temperature gradient established across a material volume at the position x at time t + τ T results in a heat flux to flow at a different instant of time t + τ q . Though it proposes a law which could be compatible with our intuition, when we adjoin it with the energy equation − ∇ q(x, t) = c (x, t), we may obtain an ill posed problem, that is, a problem which has a sequence of eigenvalues such that its real part is positive (and goes to infinity). A consequence of this fact is that the problem may be unstable and we cannot obtain continuous dependence on the initial data. 1 1 In Ignaczak (2001), found in Ref. [9 J. Ignaczak , Dual-Phase Lag Model of Rigid Heat Conductor Revisited , Proceedings of Thermal Stresses 2001, June 8–11 , 2001 , Osaka Institute of Technology , Osaka , Japan , pp. 511 – 514 . [Google Scholar]], a uniqueness theorem for arbitrary non-negative τ T and τ q was proved; together with a domain of influence theorem when τ q −τ T > 0. Similar results were also obtained in the literature described herein. These results show to what extent the dual-phase lag one-temperature theory makes sense. In this note we combine this constitutive equation with a two temperatures heat conduction theory and we show in this new context the problem is well posed. We also show that whenever the constitutive constants satisfy suitable relations, the real part of a point spectrum of the problem is negative. We also discuss a 2-temperature thermoelasticity and a higher-order theory corresponding to the higher order Taylor expansions of the fields involved.