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

TL;DR

This paper investigates the qualitative behavior of solutions in a heat conduction model with dual-phase-lag, analyzing how delay times affect solution uniqueness, stability, and spatial properties in transient and steady states.

Contribution

It provides new theoretical results on the influence of delay times on solution properties, including uniqueness, stability, and spatial behavior, in the dual-phase-lag heat conduction model.

Findings

Solutions are unique under positive definite conductivity tensor.

Solutions are stable for delay times with 0 ≤ τ_q ≤ 2τ_T.

Solutions may grow exponentially when 0 < 2τ_T < τ_q.

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.