Some Non-existence Results for the Semilinear Schrödinger Equation without Gauge Invariance

We consider the Cauchy problem for the semilinear Schrödinger equation (NLS) { (i∂t +∆)u = μ|u|, (t, x) ∈ [0, Tλ)× R, u(0, x) = λf(x), x ∈ R, where u = u(t, x) is a C-valued unknown function, μ ∈ C\{0}, p > 1, λ ≥ 0, f = f(x) is a C-valued given function and Tλ is a maximal existence time of the solution. Our first aim in the present paper is to prove a large data blow-up result for (NLS) in H-critical or H-subcritical case p ≤ ps := 1 + 4/(d− 2s), for some s ≥ 0. More precisely, we show that in the case 1 < p ≤ ps, for a suitable H-function f , there exists λ0 > 0 such that for any λ > λ0, the following estimates Tλ ≤ Cλ−κ and { limt→Tλ−0 ∥u(t)∥Hs = ∞, (if 1 < p < ps), ∥u∥Lrt (0,Tλ;B ρ,2) = ∞, (if p = ps), hold, where κ,C > 0 are constants independent of λ and (r, ρ) is an admissible pair (see Theorem 2.3). Our second aim is to prove non-existence local weak-solution for (NLS) in the Hsupercritical case p > ps, which means that if p > ps, then there exists a H -function f such that if there exist T > 0 and a weak-solution u to (NLS) on [0, T ), then λ = 0 (see Theorem 2.5).