| This dissertation is devoted to study the global well-posedness and scattering theory of the nonlinear fourth-order Schrodinger equations by the perturbation ar-gument in [6,12] and concern the local wellposedness for fourth-order Schrodinger equations by the [k,Ζ]-multiplier norm argument of Tao[39].In chapter 1, first, we take Schrodinger equation for example to state the concept of the scattering theory. Second, we introduce the definition and some properties of the Bourgain spaces.In chapter 2, we study the global well-posedness and scattering theory of the nonlinear fourth-order Schrodinger equation. Fourth-order Schrodinger equations have been introduced by Karpman and Karpman and Shagalov [19,20] to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. Our main idea is the following:first, we utilize the perturbation argument to show the "good local well-posedness", which means the time interval on which we have a well-posed solution depends only on the Hx2 norm of the initial data, rather than the profile of the initial data. This good local well-posedness combining with the global kinetic energy control yields global well-posedness. When all nonlinearities are defocusing, we get the global space-time estimate by the interaction Morawetz inequality which has appeared in [30,36] to give the scattering result. If the subcritical term is focusing but the mass is small, since when the mass is small, the subcritical term will also be small (in certain norms defined later), therefore we can utilize the perturbation argument to get global space-time estimate for the solution u of (2.1) which implies the scattering results.In chapter 3, first, we reduce the problem of local well-posedness to bilinear estimates by the fixed point theory argument. Second, we introduce the [k;Ζ]-multiplier norm method of Tao[39] and then we prove the bilinear estimates. |
PDF Full Text Request