Abstract
We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem.
Highlights
In our study, we approach a thermoelastic body having a dipolar structure
We want to enumerate some of these [9,10,11,12,13,14,15,16]: The first result for the backward in time problem belongs to Serrin, who approached this problem in the context of Navier–Stokes equations
To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem
Summary
We approach a thermoelastic body having a dipolar structure. This kind of structure falls within a more general theory, namely, the theory of bodies with microstructure. The forward in time problem, in the context of theory for thermomicrostretch elastic solids, was approached by Passarella and Tibullo in [29]. It is worth noting that the idea of considering non-standard problems, in the context of the general theory of bodies having a dipolar structure, was inspired by Quintanilla and Straughan’s work [30]. In the last part of our study we prove the main result, namely, the continuous dependence of solutions—with regards to the coefficients that couple the equations describing the dipolar deformation—with the equations that describe the behavior of voids. The description of the continuous dependence was possible due to the definition of an adequate measure
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