On The Global Wellposedness And Blow-up Of Several Nonlinear Schr(o|¨)dinger Equations

Nonlinear Schr(o|¨)dinger equation (NLS) has received much attention. Since1970s,great progress has been made after introducing harmonic analysis technique. Sevaralfamous mathematicians, such as J. Bourgain and T. Tao (both are Fields medalists), C.Kenig (Plenary speaker of International Congress of Mathematicians), F. Merle, havedone a lot of important research and made great contributions.Global wellposedness and blow-up analysis are two basic problems of NLS.Roughly speaking, global wellposedness is related to the long time existence of the solu-tions. The usual approaches include continuity arguments, conservation laws,“I method”etc.. Blow-up analysis aims to investigate the behavior of the finite time blow-up solutionsnear the blow-up time. The usual approaches include concentration compactness argu-ment, spectral analysis,“profile decomposition”, etc.. NLS with diferent type nonlinearterms, will behave diferently in global wellposedness and blow-up. When the nonlinearterm is power type, many deep results have been proved; while the nonlinear terms are ofother types, research and results are still limited.This thesis will study the following three Schr(o|¨)dinger equations: the two dimen-sional generalized Gross-Pitaevskii equation, the Gross-Pitaevskii equation with trappedDipolar gases and the mass critical fourth-order Schr(o|¨)dinger equation in higher dimen-sions. According to the speciality of each equation, suitable techniques are applied andnew results are obtained. The contributions of this thesis include:Based on Banach fixed point theorem and conservation laws, the two dimensionalgeneralized Gross-Pitaevskii equation is proved to enjoy global wellposedness in aspace bigger than H1(R2), which extends the well-known result in literature;Through constructing proper variational problem, two novel invariant sets of theGross-Pitaevskii with trapped Dipolar gases are obtained. Hence, new sharp thresh-old of global wellposedness and blow-up is proposed;The finite time blow-up solutions for the mass critical fourth-order Schr(o|¨)dingerequation in Hs(Rd)(s0(d)<s <2) concentrate not less than the mass of the groundstate, which provides new findings in this area.