Some Problems To Nonlinear Evolution Equations Of Higher Order

In this paper we study the existence uniquness and the asymptotic behavior of global solutions and the nonexistence of the global solutions for the initial boundary value problems, Cauchy problem and abstract Cauchy problem to some nonlinear evolution equations of higer order. The main results include the following five parts:In Chapter 2, we consider the initial boundary value probelm for the nonlinear wave equationwhich comes from the elasto-plastic-microstructure models. The existence of global weak solution is proved by potential well method. Also the existence and uniqueness of the generalized global solution and the classical global solution are proved.In Chapter 3, we study Cauchy problem of a class of abstract nonlinear systems with damping:The existence and non-existence of the global solution are proved by potential well method. In particular the energy decay of the solution is obtained by the use of Komornik's integral inequality.In Chapter 4, we discuss the asymptotic behavior of the weak solution for the initial boundary value problem of the following Love equationwhere fi is a bounded region in Rn with smooth boundary. The equation (5) describes the extensional vibrations of thin rods. This method is different fromthe method of Nakao difference inequality and the method of Zuazua's Ljapunov method.In Chapter 5,we discuss the Cauchy problem of the IMBq equationswhich originate from the longitudinal propagation in a nonlinear elastic rod model of DNA. We reduce the Cauchy problem of equations (8),(9) to an equivalent integral equations by the fundamental solution of a second order partial differential equation. Then using the contraction mapping principle and the extension theorem of the solution we prove the existence and uniqueness of the global generalized solutions and the existence and uniqness of the global classical solution.In Chapter 6, we investigate the generalized IMBq equations (8), (9) with the following initial boundary value condition:We reduce the problem (8)-(12) to a boundary value problem of a system of integral equations by the Green's function of a boundary value problem for a second order ordinary differential equation. Then using the contraction mapping principle and the extension theorem of the solution, we prove the existence and uniqueness of the global generalized solution and the global classical solution. We also give sufficient conditions of the blow-up of the solution by Jensen's inequality.