Existence And Decay Of Solutions For A Transmission Problem With Viscoelastic Damping

With the rapid development of modern science and technology, there have been a lot of material that has the characteristic of viscoelasticity in nature and engineering practice. Therefore, viscoelastic materials are widely used in engineering practice and life production. Transmission problems arise in several applications of physics and biology. The problem is related to the wave propagation over a body which consists of two different type of materials. In recent years, more and more scholars have paid attention to and studied the kind of problem. In this work, we investigate the existence and general decay of solution for transmission problems with viscoelastic damping. The thesis is organized as follows:In the first Chapter, we review the background and some development of the related problems and briefly describe the main work of the present thesis.In Chapter 2, we study the general decay rate for a one-dimensional transmission problem with a past history and a frictional damping and a delay term. Firstly, we prove the well-posedness result by using the semigroup theory. Then by introducing suitable energy functional and Lyapunov functional, under a hypothesis between the weight of the delay term and the the weight of the friction damping term, we establish a general decay rate result for the energy, of which the usual exponential and polynomial decay rates are only special cases.Chapter 3, we investigate a transmission problem in a bounded domain with a viscoelastic term and a delay term. we prove the well-posedness result by using Faedo-Galerkin method. Then by introducing suitable energy functional and Lyapunov func-tional, we establish a general decay rate result for the energy, of which the usual expo-nential and polynomial decay rates are only special cases.In Chapter 4, we continue to study the similar problem to Chapter 3 but without any frictional damping term. The main difficulty here is that we have no frictional damping term to control the delay term in the estimate of the energy decay. Inspired by [15,22], our idea is that use the viscoelasticity term to control the delay term. Thus, to overcome the difficulty, we need a restriction on the weight of the delay term and a new Lyapunov functional.