Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction |
| |
Institution: | 1. Faculty of Mathematics, Al. I. Cuza University of Iaşi, 700506 – Iaşi, & Octav Mayer Mathematics Institute, Romanian Academy, 700505 - Iaşi, Romania;2. Dipartimento di Ingegneria Industriale/DIIN, University of Salerno, 84084 Fisciano (SA), Italia;3. Dipartimento di Matematica, University of Salerno, 84084 Fisciano (SA), Italia |
| |
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 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. |
| |
Keywords: | |
本文献已被 ScienceDirect 等数据库收录! |
|