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

more »

... temperatures 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.