Well-posedness And Asymptotic Stability To Beam Systems Of Type ?

The thermoelastic beam equation is a mathematical model,which is established by the beam's deformation rule,structure rule,and temperature distribution rule.This kind of models are used in various fields of natural science and also have practical research background.These study of thermoelastic beam equations are always based on the study of individual equations.These equations describe the vibrational state of the beam or rod over time in a straight plane and how the future state of motion depends on the current state.In this work,we investigate the existence and decay results of soultions to thermoelastic system of type ?.The main content of this paper is as follows:In Chapter 1,we introduce the background and some development of the ther-moelastic beam system and briefly describe the main work of the present thesis.In Chapter 2,we study the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type ?.By using semigroup method and Lumer-Philips theorem,we prove the existence and uniqueness of the solution.By using the perturbed energy method and construct some Lyapunov functionals,we then obtain the exponential decay result for the case of equal wave speeds,i.e.,G/?1=D/I?1.When G/?1?D/I?1,we obtain the lack of exponential stability by using Gearhart-Herbst-Pruss-Huang theorem.For this case,by introducing the extra second-order energy,we prove the polynomial decay result.In chapter 3,by using the method of observing inequalities,we consider a one-dimensional linear Timoshenko system of type ?.By introducing some a priori esti-mates,we prove that the energy is exponentially stable under the equal wave speeds assumption and polynomial stable at non-equal wave speeds case.Moreover,by using of cut-off function,we conclude that the decay rate is independent of the boundary condition.In chapter 4,we give a summary of this work and introduce the prospect of future research.