We consider the mixed initial-boundary value problem in the context of the
Moore-Gibson-Thompson theory of thermoelasticity for dipolar bodies. We consider the case of heat conduction with dissipation. Even if the elasticity tensors
are not supposed to be positively defined, we have proven both, the uniqueness
and the instability of the solution of the mixed problem. In the case that the mass
density and the thermal conductivity tensor are positive, we obtain the uniqueness
of the solution using some Lagrange type identities.
Our study is dedicated to a composite, which, in fact, is a mixture of two thermoelastic micropolar bodies. We formulate the mixed initial boundary value problem in this context and define the domain of influence for given data. For any solution of the mixed problem we associate a measure and prove a second-order differential inequality for it. Based on the maximum principle for the heat equation and on the second-order differential inequality, we establish an estimate which proves that the thermal and the mechanical effects, at large distance from the domain of influence, are dominated by an exponential decay.
In our study, we consider the linear mixed initial boundary value problem for a porous elastic body having a dipolar structure. The equations that describe the elastic dipolar deformations are coupled with the equations which describe the evolution of the voids by means of certain coefficients. Our main result proves the continuous dependence of solutions for the mixed problem with regard to the coefficients which perform this coupling. Using an adequate measure, we can evaluate the continuous dependence by means of some estimate regarding the gradient of deformations and the gradient of the function that describes the evolution of the voids.