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

We consider the following Cauchy problem: iu(t) = Delta u del(x)u vertical bar f(x, vertical bar u vertical bar(2))u vertical bar (W(x) star vertical bar u vertical bar(2))u, x subset of R-N, t > 0, u(x, 0) = u(0)(x), x subset of R-N, where V(x) and W(x) are real-valued potentials and V(x) >= 0 and W(x) is even, f(x, vertical bar u vertical bar(2)) is measurable in x and continuous in vertical bar u vertical bar(2), and u(0)(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.