Dual-phase-lag one-dimensional thermo-porous-elasticity with microtemperatures

Z Liu,R Quintanilla

doi:10.1007/s40314-021-01603-6

Abstract

This paper is devoted to studying the linear system of partial differential equations modelling a one-dimensional thermo-porous-elastic problem with microtemperatures in the context of the dual-phase-lag heat conduction. Existence, uniqueness, and exponential decay of solutions are proved. Polynomial stability is also obtained in the case that the relaxation parameters satisfy a certain equality. Our arguments are based on the theory of semigroups of linear operators.

Highlights

At the beginning of the last century, Cosserat brothers (Cosserat and Cosserat 1909) proposed the study of the micropolar elastic solids; that is a type of material such that the material points can rotate and, microstructure is taken into account
It is worth recalling that this theory has received a lots of attention in the past years (Feng and Apalara 2019; Feng and Yin 2019; Leseduarte et al 2010; Magaña and Quintanilla 2006, 2007; Miranville and Quintanilla 2019, 2020; Pamplona et al 2011; Santos et al 2017) to understand the relevance of the microstructural component in the whole material
Liu et al (2020) proposed the study of the dual-phase-lag heat conduction with microtemperatures and gave sufficient conditions to guarantee the stability of the problem (Borgmeyer et al 2014; Liu et al 2017; Liu and Quintanilla 2018; Magaña and Quintanilla 2018)

Summary

Introduction

At the beginning of the last century, Cosserat brothers (Cosserat and Cosserat 1909) proposed the study of the micropolar elastic solids; that is a type of material such that the material points can rotate and, microstructure is taken into account. Liu et al (2020) proposed the study of the dual-phase-lag heat conduction with microtemperatures and gave sufficient conditions to guarantee the stability of the problem (Borgmeyer et al 2014; Liu et al 2017; Liu and Quintanilla 2018; Magaña and Quintanilla 2018). In this short note, we want to focus our attention to porous-thermo-elastic materials with microtemperatures in the context of the dual-phase-lag theory. We obtain the polynomial stability of the solutions when the relaxation parameters satisfy a certain relation (limit case)

Basic equations

Existence and uniqueness

Some comments