Continuous dependence and convergence for a Moore-Gibson-Thompson thermoelastic problem

In this article, we investigate how the solutions of the Moore-Gibson-Thompson thermoelasticity vary after a change of the relaxation parameter or the conductivity rate parameter, although, in the second case, only for radial solutions. The results focus on the structural stability. We also obtain the convergence of the Moore-Gibson-Thompson thermoelasticity to the type III thermoelasticity and the convergence of the Moore-Gibson-Thompson thermoelasticity to the Lord-Shulman thermoelasticity in the case of radial solutions. For the structural stability results, a certain measure for the difference of solutions can be used to control by an expression depending on the square of the difference of the parameters, and, for the convergence results, a measure of the difference of the solutions is proved to be controlled by the square of the vanishing parameter. In the proof of the above results, the energy arguments are used. It is worth saying that there are no results of this kind for the Moore-Gibson-Thompson thermoelasticity. Therefore, our results are the first contributions in this sense.