| In this thesis,the stabilization of the transmission problem of wave equations with dynamic boundary conditions is studied.The main research content is to consider the energy decay of the system by imposing velocity feedback on the dynamic boundary.The energy decay rate of the system is further considered when time delay occurs in the system and the main part of the equation is the variable coefficient.The content is mainly divided into six chapters:Chapter 1 is the introduction,which introduces the research background,research status at home and abroad,research problems and main results.In addition,some basic knowledge that will be used in this paper is also introduced.In Chapter 2,we study the stabilization of the transmission problem of wave equations with linear dynamic boundary conditions.The energy decay of the system is obtained by imposing a suitable velocity feedback on the dynamic boundary.First,the semigroup theory is used to prove that the well-posedness of the system;Afterwards,the energy estimation formula is obtained by constructing auxiliary functions and using the multiplier technique,and then exponential decay result is obtained.In Chapter 3,we focus on the stabilization of the transmission problem of wave equations with a delay in the dynamic boundary,and discuss the effect of distributed feedback control on the transmission boundary on the system decay results.Two cases are considered in this part:in the first case,velocity feedback is applied only on the boundary of the system.By constructing a suitable energy function,combined with appropriate multiplier skills and the method to deal with the delay problems,It is proved that the system is exponential decay in the case of certain restrictions on the transmission speed and transmission boundary.In the second case,not only velocity feedback is applied on the boundary,but also distribution feedback control is applied near the transmission boundary.By constructing a truncation function,the transmission term is processed to obtain the energy estimation formula,which proves that the system decays exponentially without relying on the transmission speed and transmission boundary.In Chapter 4,the stabilization of the transmission problem of wave equations with variable coefficients and interior delay is mainly discussed.Compared with the constant coefficients system,the variable coefficients system is closer to the actual situation in natural sciences and has a wider range of applications.By constructing a suitable velocity feedback control,combined with the Riemannian geometry method,the multiplier technique and the Lyapunov method,we prove that the system is exponential decay when the coefficient of the delay term is sufficiently small.In Chapter 5,we mainly study the stabilization of the transmission problem of wave equations with variable coefficients and nonlinear dynamic boundary conditions.Different from the feedback control considered in chapters 2,3 and 4,the boundary feedback control does not include the angular velocity term in the system,and the influence of the low-order term is also considered.Firstly,the low-order term is processed by using the Riemannian geometry method,the multiplier technique and the Sobolev embedding theorem,and the energy estimation formula is obtained.Secondly,a higher-order energy function is constructed to deal with the acceleration term.Finally,the explicit energy decay rate of the system is obtained by using the properties of the convex function.Chapter 6 is a summary of the results and innovations of this paper,and suggests some further questions based on the discussion in this paper. |
