Uniqueness, continuous dependence, and spatial behavior of the solution in linear porous thermoelasticity with two relaxation times

Abstract

This work aims to contribute to the verification of the well-posedness question, as for the uniqueness and continuous dependence issues, for a linear thermoelastic model with two main features: (i) a porous material matrix modeled on the basis of the Cowin–Nunziato theory; (ii) a heat transfer process obeying a time-differential constitutive equation with two relaxation times, derived from the dual-phase lag theory with an appropriate selection of the Taylor series expansion orders. Imagining to deal with very small spatial scales (in the order of the micro or nanometer), we assume it is reasonable to accept that the deformations caused by temperature variations are small enough to be realistically modeled under hypotheses of linearity, thus making the mathematical theory under investigation particularly meaningful, e.g. in the study of miniaturized devices in very fast transients. The work is concluded by proving a domain of influence theorem, and with a reference to the future research activities to carry out starting from it.

Keywords:

• Continuous dependence
• domain of influence
• delay times
• heat conduction models
• porous media
• Taylor series expansion
• uniqueness

Acknowledgments

V. Z. acknowledges the Italian National Group of Mathematical Physics (GNFM-INdAM) for supporting the research project Progetto Giovani 2018 Heat-pulse propagation in FGMs; A. A. acknowledges financial support from the Italian Ministry of Education, University and Research (MIUR) under the Departments of Excellence grant L. 232/2016. The authors also wish to thank the anonymous reviewer for his/her valuable contribution to the improvement of the article.