L~2-concentration Of Blow-up Solutions For Nonlinear Schr(?)dinger Equation

In this paper, we make investigations on the dynamical propertiesof blow-up solutions for the nonlinear schr(o|¨)dinger equation with potential inthe energy space.First, we consider the Cauchy problem of the cubic nonlinear schr(o|¨)dingerequation with potential in two dimensional spaces. By defining the functional,the sufficient condition for the solutions blowing up in finite time, nonexistenceof strong limit in L~2 as t→T(T is the blow-up time) and L~2-concentration ofblow-up solutions are proved.Furthermore, the above results are generalized to the critical nonlinearschr(o|¨)dinger equation with potential in RN.Lastly, we studied the Cauchy problem of the nonlinear schr(o|¨)dinger equa-tion with harmonic potential. The similarly, some properties are proved: thesufficient condition for the solutions blow up in finite time, nonexistence ofstrong limit in L~2 as t→T(T is the blow-up time) and L~2-concentration ofblow-up solutions.