Some Non-Standard Problems Related with the Mathematical Model of Thermoelasticity with Microtemperatures

This article considers the linear theory of thermoelastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. We discuss some non-standard problems within the context of the dynamic boundary value problems. In fact, we use the Lagrange-Brun type identities combined with some differential inequalities in order to show that the final boundary value problem associated with the linear thermoelasticity with microtemperatures has at most one solution in appropriate classes of displacement-temperature-microtemperature fields. Furthermore, we study the constrained motion of a prismatic cylinder made of a thermoelastic material with microtemperatures and subjected to final given data that are proportional, but not identical with, the respective values at a prescribed early time. We show that certain cross-sectional integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived upon the proportionality coefficients in order to show that the cross-sectional integrals exhibit alternative behavior and, in particular for a semi-infinite cylinder, that there is either at least exponential growth or at most exponential decay.