Global Wellposedness And Scattering For Some Combined Nonlinear Schr(?)dinger Equation In The Radial Case

In this thesis, we mainly study the global well-posedness and scattering versus blow-up for a class of nonlinear Schrodinger equation with combined power-type non-linearities in the radial case. We also consider an approximate problem. The thesis is organized as follows.In the introduction, the origin, developments and main contents of nonlinear Schrodinger equation, are briefly recalled.In Chapter2, we mainly recall some basic notions and properties in nonlinear Schrodinger equation and harmonic analysis which will be used in this thesis.In Chapter,3, we study the global well-posedness and scattering versus blow-up for the nonlinear Schrodinger equation with combined power-type nonlinearities in the radial case. More precisely, We will consider a combined nonlinear Schrodinger equa-tion with the nonlinearity be a mass-critical defocusing term plus a mass supercritical and also energy subcritical or energy critical focusing term in the radial case, that is where1+4/d<p<∞, d=1,2and1+4/d<p≤1+4/d-2,d=3,4. We show the global well-posedness and scattering versus blow-up dichotomy when the Lyapunov functional of the initial data u0below the Lyapunov functional of the ground state in the radial case. First, by using variational estimate, we give the ground state. Then we use the scaling derivation (?) of the Lyapunov functional to define the scattering set and the blow-up set. That is, when the Lyapunov functional of the initial data u0below the Lyapunov functional of the ground state, if (?)(u0)≥0, the equation is global well-posedness and scattering, while for (?)(u0)<0, the solution will blow-up in finite time. To show the scattering theory, We develop a new profile decomposition in the energy space by using the profile decomposition in the mass-critical case. By using this profile decomposition together with the concentration compactness method, we establish the global well-posedness and scattering. For the blow-up case, we prove the finite time blow-up by using the virial identity.In Chapter4, we consider an approximate problem. We will consider the follow- ing Cauchy problem We study the behavior of the equation in the radial case when ε tends to0. We show that for some radial initial data φεâ†'φ in H1(Rd), the solution uε{t) tends to ν(t) in H1(Rd) for any time t∈Rî–“, where ν(t) is the solution of the mass critical defocusing nonlinear Schrodinger equation with initial data φ.