Small data finite-time blow-up solutions for the nonlinear Schrödinger equation in general dimensions

The purpose of this paper is to show the existence of finite-time blow-up solution for the nonlinear Schrödinger equation in general spatial dimension which starts with the initial data as small as we like. We construct a solution of the nonlinear Schrödinger equation defined in negative global time and blows-up at positive finite time. We derive decay estimates for the solutions in the negative time direction by using interpolation inequalities in the weighted Lebesgue spaces.