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

Abstract

Abstract Three dual-phase-lag heat conduction theory is based on the constitutive law q ( P , t + τ q ) = − ( k ∇ T ( P , t + τ T ) + k ∗ ∇ ν ( P , t + τ ν ) ) , ( 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 ) = c T ˙ ( 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 τ ν > τ q = τ T . 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.