Some solutions for a family of exact phase-lag heat conduction problems

Three dual-phase-lag heat conduction theory is based on the constitutive law q(P,t+τq)=−(k∇T(P,t+τT)+k∗∇ν(P,t+τν)),(ν˙=T). It is an extension of the dual-phase-lag which is able to recover the Green and Nagdhi theories when Taylor approximations are considered. If we adjoin this constitutive law with the energy equation −∇q(x,t)=cT˙(x,t), an ill-posed problem is generically obtained. That is, a problem with a sequence of eigenvalues for which the real part goes to infinity. As a consequence, the problem is unstable and, moreover, there is no continuous dependence of solutions with respect initial data. In this note we show that this behavior does not apply when τν > τq = τT. We prove continuous dependence with respect the initial data and supply terms. We also show how to obtain the solutions of the problem by means of a recurrent scheme. Travelling wave solutions are also obtained. The continuous dependence results are extended to the thermoelastic case and several approximation theories are considered.

► Well-posedness for an exact phase-lag equation. ► Description of the solutions of exact phase-lag equation. ► A phase-lag thermoelastic problem. ► Taylor approximations top the phase-lag equation.