On A Kind Of Fourth Order Nonlinear Schr(?)dinger Equations

The initial-boundary value problem for the following fourth order nonlinear Schr(?)dinger equation (?) is studied in this paper,where a is a positive constant, u is complex function, and ?(?)R4 is a bounded domain with sufficiently smooth boundary. The system describes the propagation models of intense laser beam in a bulk medium with Kerr nonlinearity. The initial-boundary value problem of the above equation is studied. For the case of a ? 1, approximate solution'method is used in the paper. Firstly the approximate solution sequence is constructed by using the smooth operator. Secondly, the uniform estimate of approximate solution is established. Lastly, limit of approximate solution sequence is just the solution of the original equation is proved. The main skill of the proof is to estimate the higher order norm of the solution by means of taking one derivation in the equation about time. Especially, for the case of 1 ? a ? 3 , Priori estimates of the solutions are established by means of energy estimates and improved B-G type inequalities. The global existence and uniqueness of the solution are obtained by using Galerkin's method.