Some remarks on growth and uniqueness in thermoelasticity

Abstract

We use the Lagrange identity method and the logarithmic convexity to obtain uniqueness and exponential growth of solutions in the thermoelasticity of type III and thermoelasticity without energy dissipation. As this is not the first contribution of this kind in this theory, it is worth remarking that the assumptions we use here are different from those used in other previous contributions. We assume that the elasticity tensor is positive semidefinite, but we allow that the constitutive tensor of the entropy flux vector (k ij ), which is a characteristic tensor in this theory, is not sign-definite. The Lagrange identity method is used to obtain uniqueness in the context of the thermoelasticity of type III. The fundamental key to obtain exponential growth in the thermoelasticity without energy dissipation is the use of a new functional. This functional is inspired in that it is used when the elasticity tensor is not sign-definite, but (k ij ) is positive definite.

On the dynamical and static problems in magneto-elasticity

“…This instability result is similar to that obtained in [11][12][13] in thermoelasticity without energy dissipation and thermoelasticity of type III.…”

On uniqueness for a family of nonstandard problems

“…When either a il jk is positive definite or k ik is positive definite we can obtain also uniqueness of solutions whenever 11) and the nonstandard initial conditions…”

Global existence for the three-dimensional thermoelastic equations of Type II

“…Racke, Shibata and Zheng [29] obtained the global existence and uniqueness of solutions for the nonlinear thermoelastic system of type I with small initial data; Muñoz Rivera and Qin [18] proved the global existence, uniqueness, and asymptotic behavior of solutions for 1D nonlinear thermoelasticity with thermal memory subject to Dirichlet-Dirichlet boundary conditions. For the thermoelasticitic model of type II, or without energy dissipation, several results on existence, uniqueness, continuous dependence, spatial decay and wave propagation (see, e.g., [4]- [5], [10,15,19], [24]- [27]) have been obtained, among which we would like to mention especially the work by Qin and Muñoz Rivera [24], who studied the global existence and exponential stability of solutions to homogeneous thermoelastic equations of type II with thermal memory. Recently, Qin, Xu and Ma [25] obtained the global existence and exponential stability of solutions to nonhomogeneous thermoelastic equations of type II with thermal memory.…”