Existence and exponential decay in nonlinear thermoelasticity (original) (raw)
Related papers
Decay Rates for the Three-Dimensional Linear System of Thermoelasticity
Archive for Rational Mechanics and Analysis, 1999
We consider the two and three-dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as t → ∞. First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lamé system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component. We show that when the domain is convex, the decay rate is never uniform. In fact, the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents. We also give an example, due to D. Hulin, in which these geometric properties hold. In three space dimensions we indicate (without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite-dimensional subspace of solutions except when the domain is a ball or an annulus. Finally we discuss some closely related controllability and spectral issues.
Exponential Decay to Partially Thermoelastic Materials
Sunto.-Studiamo il sistema termoelastico per materiali che siano parzialmente termoelastici. Consideriamo cioè un materiale diviso in due parti, una delle quali sia un buon conduttore di calore, in modo che ivi esistano fenomeni termoelastici. L'altra parte materiale è un cattivo conduttore di calore e quindi non esiste il flusso di calore. In questo lavoro dimostriamo che per tali modelli la soluzione decade esponenzialmente a zero quando il tempo tende all'infinito. Studiamo anche il caso non lineare. Summary.-We study the thermoelastic system for material which are partially thermoelastic. That is, a material divided into two parts, one of them a good conductor of heat, so there exists a thermoelastic phenomenon. The other is a bad conductor of heat so there is not heat flux. We prove for such models that the solution decays exponentially as time goes to infinity. We also consider a nonlinear case.
Zeitschrift für angewandte Mathematik und Physik, 2012
Your article is protected by copyright and all rights are held exclusively by Springer Basel AG. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your work, please use the accepted author's version for posting to your own website or your institution's repository. You may further deposit the accepted author's version on a funder's repository at a funder's request, provided it is not made publicly available until 12 months after publication.
Exponential stability in one-dimensional non-linear thermoelasticity with second sound
Mathematical Methods in the Applied Sciences, 2004
In this paper, we consider a one-dimensional non-linear system of thermoelasticity with second sound. We establish an exponential decay result for solutions with small 'enough' initial data. This work extends the result of Racke (Math. Methods Appl. Sci. 2002; 25:409 -441) to a more general situation.
submitted for publication
In this paper we prove the global existence and exponential stability of solutions to thermoelastic equations of hyperbolic type provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially. Moreover, the global solution, together with its the third-order full energy, is exponentially stable for any t > 0.