Well-posedness And Decay Of A Laminated Beam With Structural Damping

With the rapid development of science and technology, multi beam structures have been widely used in engineering practice. Therefore, in practical applications,we often see the structure of the same two beams through a thin and light adhesive layer bonded together, such a structure is called laminated beam. Laminated beam problem has been more and more widely used in the field of mechanical engineering,civil engineering and aerospace. In this work, we investigate the well-posedness and decay results for laminated beam with structural damping and viscoelastic damping or the heat dissipation. The main contents are 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 for a laminated beam with structural damping and viscoelastic damping. First of all, in the literature [26], Lo and Tatar obtained the exponential decay result in the case of equal speeds of wave propagation.In the case of equal and non-equal speeds of wave propagation, by constructing the Lyapunov functional related to the energy functional and using the second-order energy method, we prove the general decay result, respectively.In Chapter 3, we investigate a laminated beam with structural damping and second sound. First of all, in the literature [2], Apalara studied the laminated beam with structural damping and second sound. The author introduced a stability number which is related to the coefficients of problem. Exponential decay and polynomial decay are obtained respectively when the stability number is 0 and not 0. In this chapter, we solve the open problem in the literature [2]. That is, when the stability number is not 0,the solution is non-exponential decay. We prove the result by using the Pruss theorem.In Chapter 4,we continue to study a similar problem to that of Chapter 3 but with an additional delay term, that is to study the laminated beam with structural damping, second sound and time delay. First of all, we prove the well-posedness of solutions by using the semigroup method. Then, under the condition that the delay coefficient satisfies certain conditions, we introduce the same stability number as in the chapter 3, when the stability number is equal to 0, we prove the exponential decay of the solution by constructing the Lyapunov functional equivalent to the energy functional.The main difficulty here is that the additional delay term may lead to instability of the system. Inspired by [3], our idea is to use the second sound to control the delay term.