A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation

Abstract

Abstract. In this paper we consider the nonlinear Schrödinger equation iut + ∆u + κ|u| α u = 0. We prove that if α < 2 N and ℑκ < 0, then every nontrivial H 1 -solution blows up in finite or infinite time. In the case α > 2 N and κ ∈ C, we improve the existing low energy scattering results in dimensions N ≥ 7. More precisely, we prove that if, then small data give rise to global, scattering solutions in H 1 .

Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method

“…For the complex Ginburg-Landau equation without gauge invariance (1.5), blowup and lifespan of solutions to (1.5) in one-dimensional torus is studied by Ozawa-Yamazaki [38]. Of course the complex Ginburg-Landau equation with nonlinear term (κ + iβ)|u| p−1 u (with gauge invariance) has been considered (see for existence, e.g., Ginible-Velo [10], Okazawa-Yokota [37], and for blowup phenomena, e.g., Masmoudi-Zaag [29], Cazenave-Dickstein-Weissler [3] and Cazenave-Correia-Dickstein-Weissler [2]).…”

Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

“…Another main result of this paper concerns the finite time blowup of the fourthorder Schrödinger equation (1.1). The existence of blowup solutions for the equation (1.1) has been strongly supported by a series of numerical studies done by Fibich, Ilan and Papanicolaou [18] for mass-critical and mass-supercritical powers α ≥ 8 N . Recently, it has been proved rigorous in Boulenger [6], under the natural criteria from the well-known blowup results for the classical nonlinear Schrödinger equation, that the blowup solutions exist for the radial data in H 2 (R N ) with the negative initial energy.…”

Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

“…See [3,2]. Moreover, if α < 2 N , then all nontrivial solutions blow up in finite or infinite time, see [1]. Finite-time blowup also occurs if λ = 0, λ > 0, and α ≥ 4 N , since in this case (1.1) is the focusing NLS.…”

“…Thus we see that the asymptotic behavior of u(t) as t → ∞ is described by the function z(t) via the estimate (1.11). Note that the functions f 0 and Ψ are both real-valued, and that 1 2 ≤ 1 + f 0 ≤ 3 2 and 0 < Ψ ≤ 1. The function Θ is also real-valued.…”