Small data blow-up of solutions to nonlinear Schrodinger equations without gauge invariance in L2

Abstract

In this article we study the Cauchy problem of the nonlinear Schrödinger equations without gauge invariance i∂tu + Δu = λ(|u|p1 + |v|p2, (t, x) ∈ [0, T) x ℝn, i∂tv + Δv = λ(|u|p2 + |v|p1, (t, x) ∈ [0, T) x ℝn, where 1 < p1, p2 < 1 + 4/n and λ ∈ ℂ\{0}. We first prove the existence of a local solution with initial data in L2(ℝn). Then under a suitable condition on the initial data, we show that the L2-norm of the solution must blow up in finite time although the initial data are arbitrarily small. As a by-product, we also obtain an upper bound of the maximal existence time of the solution.