Researches Of The Global Well-posedness Of Thermoelastic System And Related Models

Thermoelastic equations describe the elastic and the thermal behavior of elastic, heat conductive media, in particular the reciprocal actions between elastic stresses and temperature differences. The present dissertation is concerned with the global exis-tence and asymptotic behavior of solutions to thermoelastic systems, thermoviscoelas-tic systems and Timoshenko systems. Moreover, the existence of a global attractor is achieved in some case.This dissertation is divided into seven chapters.Chapter 1 is preface.In Chapter 2, we prove the global existence and exponential stability of solutions to nonlinear thermoelastic equations with second sound provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.In Chapter 3, we consider the stability property for Timoshenko-type systems with past history g (the relaxation kernel). For g decaying polynomially, we prove polyno-mial stability results for the equal wave-speed propagation; for the nonequal wave-speed case, we also establish a decay result under the exponential decay condition on g. Moreover, the existence of a global attractor is achieved.In Chapter 4, we establish the global existence result for the higher-dimensional linear thermoviscoelastic equations by using a semigroup approach. Using multipler techniques and Lyapunov methods, we prove that the energy in the higher-dimensional linear thermoviscoelasticity decays to zero exponentially by introducing a velocity feed-back on a part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary to increase the loss of energy.In Chapter 5, we obtain a decay result for higher-dimensional linear thermovis- coelastic equations by introducing a velocity feedback on a part of the boundary and using the multiplier techniques method. Moreover, the existence of a global attractors is abtained.In Chapter 6, we consider a one-dimensional continuous model of nutron star, which is described by a compressible thermoviscoelastic system with a non-monotone equation of state, due to the effective Skyrme nuclear interaction between particles. We prove that, despite a possible destabilizing influence of the pressure, which is non-monotone and not always positive, the presence of viscosity and a sufficient thermal dissipation can yield the global existence of solutions in H4 with a mixed free boundary problem for our model.In Chapter 7, we summarize of the results of the dissertation, and predict the work in the future.