On Dynamical Behavior Of Nonlinear Schr(o|")dinger Equations

In this paper, we consider the dynamic behavior for the solutions to the Cauchy problems of several types of nonlinear Schodinger equations. Firstly, we consider the following Cauchy problem where V(x) and W(x) are real-valued potentials, V(x)≥0and W(x) is even, f(x,|u|2) is measurable in x and continuous in|u|2, uo(x) is a complex-valued function of x. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem. Secondly, we consider a special case of (1): where N≥3,0<p1<p>2<4/N2,μ and v are real constants. We establish two sharp thresholds for blowup and global existence of the solution when4/N<p24/N-2and discuss the instability of the standing wave solution. Finally, we consider the following Cauchy problemWe will show that any blowup solution satisfies the mass concentration phenomenon near the blowup time.