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

Abstract

We consider the Cauchy problem for the semilinear Schrödinger equationis a C-valued given function and T λ 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 s -critical or H s -subcritical case p ≤ p s := 1 + 4/(d − 2s), for some s ≥ 0. More precisely, we show that in the case 1 < p ≤ p s , for a suitable H s -function f , there exists λ 0 > 0 such that for any λ > λ 0 , the following estimateshold, 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 H s -supercritical case p > p s , which means that if p > p s , then there exists an H s -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).

A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity

“…We also remark that in the 1‐dimensional case, s > 1/2 is also the necessary condition for the local existence in the framework because the nonexistence of local weak solutions for with some data is shown in Fujiwara . In a general setting, the necessary condition is still open, and partial results are discussed in Ikeda and Inui . On the other hand, is scaling invariant.…”

“…We consider the Cauchy problem for the following semirelativistic equations with nongauge invariant power type nonlinearity: with , where ∂ t = ∂ / ∂ t and Δ is the Laplacian in . Here, (−Δ) 1/2 is realized as a Fourier multiplier with symbol | ξ |: , where is the Fourier transform defined by We remark that the Cauchy problem such as arises in various physical settings, and accordingly, semirelativistic equations are also called half‐wave equations, fractional Schrödinger equations, and so on; see previous studies and reference therein.…”

The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance

“…The results in [5][6][7] indicate the nonlinear effect of λ|u| p is very different from the gauge invariant nonlinearity µ|u| p−1 u. For the nonlinear Schrödinger equation with a gauge invariant power nonlinearity…”

“…In [6], Ikeda and Inui showed the blow-up of solution for the semilinear Schrödinger equation with small data when 1 < p < 1 + 4 N , which improved the result obtained in [5], and they also determined the critical exponent, and gave an upper bound of the life span for blowing up solution. Furthermore, Ikeda and Inui [7] got a large data blow-up result in H s (R N ) with a nonlinear term F (u) satisfying some conditions. The main tool in [5][6][7] is test function method.…”

“…Furthermore, Ikeda and Inui [7] got a large data blow-up result in H s (R N ) with a nonlinear term F (u) satisfying some conditions. The main tool in [5][6][7] is test function method. The method based on rescalings of a compactly support test function is an effective method to prove the nonexistence of weak solutions for many kinds of equations.…”