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

Abstract

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 analyse 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 $\tau_q$ and $\tau_T$ vary in the set $\{0\leq \tau_q\leq 2\tau_T\}\cup \{0<2\tau_T< \tau_q\}$. For the continuous dependence problem we establish two different estimates. The first one is obtained for the delay times with $0\leq \tau_q \leq 2\tau_T$, 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 $0<2\tau_T< \tau_q$ 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 $\left\{0<\tau_q\leq 2\tau_T\right\}\cup \left\{0<2\tau_T<\tau_q\right\}$. 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.