On The Long Time Behavior Of Some Schr(?)dinger Equations

Schr(?)dinger equation is a fundamental mathematical equation in quantum mechanics,widely used in the non-relativistic atomic physics,nuclear physics and solid physics.It describes the properties of particles in low speed,such as electrons,atoms and molecules.The thesis consists of five chapters.The first chapter introduces the basic discussed issues about long time behav-ior of Schr(?)dinger equations,such as scattering,blow-up phenomenon,stability and so on.There lists some examples in these issues.The second chapter is devoted to the scattering conjecture for the 4 dimen-sional defocusing energy-super-critical Schr(?)dinger equation,but in the radial case.We proved that if the critical Hsc norm stayed bounded in the maximal life-span,where sc>3/2,then the radial solution to the equation must be global and should scatter.The main ingredients here are the method of induction on energy,long-time Strichartz estimates and frequency localized Lin-Strauss type Morawetz estimates.Chapter three mainly describes the finite time blow-up solution for the mass-critical focusing Hartree equation with inverse-square potential with Hl initial data.As a prelude,the local theory and blow-up criteria was given through the Strichartz estimates and the harmonic tools related to inverse-square poten-tial.Then we give the description to the finite time blow-up solutions.As a by-product,the global existence for the solution with mass below the ground state is obtained.Variational method,together with profile decomposition and compactness concentration method,plays a central role here.Chapter four gives a qualitative description for the blow-up rate to the finite time blow-up solution for the inner-critical focusing nonlinear Schr(?)dinger equa-tion with inverse-square potential.We considered the E initial data and radial data separately.The main technique here is the method of convex analysis.At last,we introduce the unconditional uniqueness.It claims that the spe-cial solutions we discuss in these issues from long-time behavior would coincide with the one it naturally should be.As an example,we give the unconditional uniqueness for the fourth-order Schr(?)dinger equation in high dimension,with the help of Strichartz estimates and Bony product.