On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations

Abstract

Solutions to the Cauchy problem for the equation iut=Δu+F(|u| 2)u (x∈ℝn, t>0), u(x,0)=φ(x), are considered. Conditions on φ and F are given so that, for solutions with nonpositive energy, the following obtains: There exists a finite time T, estimable from above, such that ∥gradu(t)∥L2(ℝn)→+∞ as t→T−. It is also shown that other Lq-norms of a solution (including q=∞) blow up in finite time.