Existence And Decay Of Timoshenko System Of Type Ⅲ With Viscoelastic Damping

Recently, theories of viscoelastic beam equations paly an increasing role in many fields, such as engineering, physics and material science, due to its wide applications. In this work, we investigate the existence and decay properties of solutions to Timoshenko system of type Ⅲ with viscoelastic damping. This thesis is organized as follows:In Chapter 1, we review the background and some development that related to the Timoshenko problems and summarize the main work of the present dissertation.In Chapter 2, we consider the global existence and the general energy decay for Timoshenko system of type Ⅲ with a frictional damping and thermo-viscoelastic damp-ing and delay term in the internal feedback. Firstly, by using the Faedo-Galerkin ap-proximations together with some energy estimates, and under some restriction on the parameters the term of delay and the term of friction damping, we give the existence result of the global existence of the weak solutions. Then, under the hypothesis be-tween the weight of the delay term and the weight of the friction damping term, we prove a general decay of the total energy of our problem by introducing appropriate Lyapunov functionals from which the exponential and polynomial types of decay are only special cases.. We recall that for that the term of delay is equal to the term of friction damping, Nicaise and Pignotti showed in [1] that some instabilities may occur. Here, due to the presence of the viscoelastic damping, we can still establish a general energy decay result even if the term of delay is equal to the term of friction damping.In Chapter 3, we devote to investigate the Timoshenko system of type Ⅲ with a viscolastic term and delay term in the interior feedback but without the frictional damping term. By introducing the suitable energy and the Lyapunov functional, un-der suitable assumptions, we establish an exponential energy decay result. The main difficulty in handling this problem is that in the third equation we have no friction-al damping term to control the delay term in the estimate of the energy decay. To overcome this difficulty, our basic idea is to control the delay term by making use of the viscoelasticity term. And in order to arrive at this goal, a restriction of the size between the parameter μ and the kernel g and a suitable energy is needed.In Chapter 4, we investigate the asymptotic behavior of the non-autonomous Tim-oshenko system of type Ⅲ with a viscoelastic damping. Under some assumptions, we establish the asymptotic behavior by introducing the suitable energy and the Lya-punov functional. The main difficulty in handing this problem is that when both the viscoelastic term and the non-autonomous terms are present, then the analysis of their interaction and their influence on the asymptotic behavior of solutions becomes more difficult. Firstly, we use the idea in [2] to overcome the difficulty. Then, by adding restrictions on the coefficients and the relaxing function g and the non-autonomous terms, we show that the energy decays exponentially by using the energy method.