Blow-up and strong instability of standing waves for the NLS-δ equation on a star graph. (July 2020)

• Record Type: Journal Article
• Authors: Goloshchapova, Nataliia; Ohta, Masahito
• Is Part Of: Nonlinear analysis. Volume 196(2020)
• Publication Date: 2020-07
• Subjects: 35Q55 -- 81Q35 -- 37K40 -- 37K45 -- 47E05; δ- and δ′-interaction -- Nonlinear Schrödinger equation -- Strong instability -- Standing wave -- Star graph -- Virial identity; 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.2020.111753
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with δ -interaction on a star graph Γ . The key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem. We also show well-posedness of the corresponding Cauchy problem in the domain of the self-adjoint operator which defines δ -interaction. This permits to prove virial identity for the H 1 -solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS- δ ′ equation on the line.