Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

We consider the following Cauchy problem: -iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u, x∈ℝN,t>0, u(x, 0)=u0(x),x∈ℝN, where V(x) and W(x) are real-valued potentials and V(x)≥0 and W(x) is even, f(x,|u|2) is measurable in x and continuous in |u|2, and u0(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