Some non-existence results for the semilinear Schrödinger equation without gauge invariance

We consider the Cauchy problem for the semilinear Schrödinger equationequation(NLS){(i∂t+Δ)u=μ|u|p,(t,x)∈[0,Tλ)×Rd,u(0,x)=λf(x),x∈Rd, where u=u(t,x)u=u(t,x) is a CC-valued unknown function, μ∈C\{0}μ∈C\{0}, p>1p>1, λ≥0λ≥0, f=f(x)f=f(x) is a CC-valued given function and TλTλ 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 HsHs-critical or HsHs-subcritical case p≤ps:=1+4/(d−2s)p≤ps:=1+4/(d−2s), for some s≥0s≥0. More precisely, we show that in the case 10λ0>0 such that for any λ>λ0λ>λ0, the following estimatesTλ≤Cλ−κand{limt→Tλ−0⁡‖u(t)‖Hs=∞(if10κ,C>0 are constants independent of λ   and (r,ρ)(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 HsHs-supercritical case p>psp>ps, which means that if p>psp>ps, then there exists an HsHs-function f   such that if there exist T>0T>0 and a weak-solution u to (NLS) on [0,T)[0,T), then λ=0λ=0 (see Theorem 2.5).