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

Abstract

We consider the following Cauchy problem: $-i{u}_t=\mathrm{\Delta }u-V(x)u+f(x,{|u|}^{\mathrm{2}})u+(W(x)\mathrm{\star }{|u|}^{\mathrm{2}})u,\mathrm{}\mathrm{}\mathrm{}\mathrm{}x\in {ℝ}^N,\mathrm{}\mathrm{}\mathrm{}\mathrm{}t>\mathrm{0},u(x,\mathrm{0})=u_{\mathrm{0}}(x),\mathrm{}\mathrm{}\mathrm{}\mathrm{}x\in {ℝ}^N,$ where $V(x)$ and $W(x)$ are real-valued potentials and $V(x)\ge \mathrm{0}$ and $W(x)$ is even, $f(x,|u{|}^{\mathrm{2}})$ is measurable in $x$ and continuous in $|u{|}^{\mathrm{2}}$, and $u_{\mathrm{0}}(x)$ 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.