Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance. (September 2020)

• Record Type: Journal Article
• Authors: Inui, Takahisa; Kishimoto, Nobu; Nishimura, Kuranosuke
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: 35Q55 -- 35B44 -- 35B34; Mass-resonance -- Virial identity -- Blow-up -- Grow-up -- Radial symmetry -- Quadratic Schrödinger system; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111895
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the quadratic nonlinear Schrödinger system i ∂ t u + Δ u = v u ¯, i ∂ t v + κ Δ v = u 2, on I × R d, where 1 ≤ d ≤ 6 and κ > 0 . In the lower dimensional case d = 1, 2, 3, it is known that the H 1 -solution is global in time. On the other hand, there are finite time blow-up solutions when d = 4, 5, 6 and κ = 1 ∕ 2 . The condition of κ = 1 ∕ 2 is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when d = 5, 6 and κ ≠ 1 ∕ 2 and we show blow-up or grow-up when d = 4 .