Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction

In this paper we study the time differential dual-phase-lag model of heat conduction incorporating the microstructural interaction effect in the fast-transient process of heat transport. We analyze the influence of the delay times upon some qualitative properties of the solutions of the initial boundary value problems associated to such a model. Thus, the uniqueness results are established under the assumption that the conductivity tensor is positive definite and the delay times τq and τT vary in the set 0 ≤ τq ≤ 2τT ∪ 0 < 2τT < τq. For the continuous dependence problem we establish two different estimates. The first one is obtained for the delay times with 0 ≤ τq ≤ 2τT, which agrees with the thermodynamic restrictions on the model in concern, and the solutions are stable. The second estimate is established for the delay times with 0 < 2τT < τq and it allows the solutions to have an exponential growth in time. The spatial behavior of the transient solutions and the steady-state vibrations is also addressed. For the transient solutions we establish a theorem of influence domain, under the assumption that the delay times are in 0 < τq ≤ 2τT ∪ 0 < 2τT < τq. While for the amplitude of the harmonic vibrations we obtain an exponential decay estimate of Saint–Venant type, provided the frequency of vibration is lower than a critical value and without any restrictions upon the delay times.