Remarks on Blowing-Up of Solutions for Some Nonlinear Schrödinger Equations

Abstract

We study the blowing-up conditions of solutions for nonlinear Schrödinger equations with interaction which does not satisfy known Glassey's condition [4]. We also give some remarks on the blowing-up conditions on an exterior domain with a star-shaped complement under the Dirichlet boundary condition and on a complement of a ball under the Neumann boundary condition. Finally, we show global existence of solutions for the equation: $i\dfrac{\partial u}{\partial t}=\Delta u+\left(\dfrac{1}{|x|^2}*|u|^2\right)u$.