Well-posedness And Stability Of Solutions For A Class Of Coupled Systems

This thesis mainly studies the well-posedness and stability of the coupled wave equation and the beam equation.The structure of this paper is as follows:The first chapter is the introduction,which mainly introduces the research background and significance of this paper,the research status of the well-posedness and stability of the coupling system solution and the research methods used in this paper.The second chapter gives the relevant definitions,theorems and inequalities needed in this paper.In the third chapter,the coupled wave equations with inconsistent boundary types are discussed.The boundary of one wave equation is Dirichlet boundary,and the boundary of another wave equation is Neumann boundary.According to Sobolev embedding theorem and Lumer-Phillips theorem,the well-posedness of the coupled system solution can be obtained.By proving that the eigenvalues of the system are all in the left half plane,the system is asymptotically stable,and after further calculation,the system is not exponentially stable.In the fourth chapter,the coupled wave equations with consistent boundary types are studied.According to Sobolev embedding theorem and Lumer-Phillips theorem,the wellposedness of the coupled-wave equations can be obtained.The Lyapunov function method can obtain the exponential stability of the system.In the fifth chapter,the coupled beam equations with consistent boundary types are discussed.According to Sobolev embedding theorem and Lumer-Phillips theorem,the wellposedness of the coupled beam equations can be obtained.The Lyapunov function method shows that the system is exponentially stable.In short,for the coupled wave equation and beam equation,the semigroup theory can be used to prove the well-posedness of the system solution,and the stability of the system can be studied by spectral analysis or constructing the appropriate Lyapunov function method.