Blow-up of H¹ solution for the nonlinear Schrödinger equation

We consider the blow-up of the solution in H¹ for the following nonlinear Schrödinger equation: i ∂ ∂tu + Δu = -|u|^{p - 1}u, x ε{lunate} Rⁿ, t ≥ 0, (*) u(0, x) = u₀(X), x ε{lunate} Rⁿ, t = 0, where n ≥2 and 1 + 4/n ≤ p < min { (n + 2) (n - 2), 5}. We prove that if the initial data u₀ in H¹ are radially symmetric and have negative energy, then the solution of (*) in H¹ blows up in finite time. We do not assume that xu₀ ε{lunate} L², and therefore our result is the generalization of the results of Glassey [4] and M. Tsutsumi [18] for the radially symmetric case.