On The Blow-up Solutions For Some Schr(?)dinger Equations(Systems)

In this dissertation, we study blow-up, global existence and ground state solutions for some kinds of focusing nonlinear Schrodinger equations (systems).In Chapter1, we discuss the physical background of Schrodinger equa-tion(system), and list the main results obtained in this thesis. We also state some preliminaries which may be applied to the remaining chapters throughout the disser-tation.In Chapter2, we show the blow-up threshold for nonlinear Schrodinger equation with combined terms in the energy space h1(Rd), In the L2supercritical case, we give a sharp blow-up threshold which is given by the energy and the Virial functional. In the L2critical case, we give a "rough" characterization of the blow-up behavior.In Chapter3, we consider the blow-up solutions for N-coupled focusing nonlin-ear Schrodinger system in R. Firstly, we establish the ground state solutions. Sec-ondly, we derive finite time blow-up solutions under certain conditions. Finally, we verify that any solution with L2-critical nonlinearity satisfies a mass concentration phenomenon near the blow-up time.In Chapter4, we study blow-up, global existence and ground state solutions for N-coupled focusing nonlinear Schrodinger system in Rd (d≥3). Firstly, using the Nehari manifold approach and some variational techniques, the existence of ground state solutions is established. Secondly, under certain conditions, finite time blow-up phenomena of the soluions is derived. Finally, by introducing a refined version of compactness lemma, the L2concentration for the blow-up solutions is obtained.