Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 4

Threshold solutions for the 3D focusing cubic-quintic nonlinear Schrödinger equation at low frequencies

Pages: 263 – 297

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n4.a1

Authors

Masaru Hamano (Faculty of Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan)

Hiroaki Kikuchi (Department of Mathematics, Tsuda University, Kodaira-shi, Tokyo, Japan)

Minami Watanabe (Graduate School of Mathematics, Tsuda University, Kodaira-shi, Tokyo, Japan)

Abstract

This paper addresses the focusing cubic-quintic nonlinear Schrödinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the spirits of Duyckaerts and Merle $\href{ https://doi.org/10.1007/s00039-009-0707-x}{[14]}$. When we try to obtain the corresponding results of [14], we meet several difficulties due to the cubic-quintic nonlinearity. We overcome them by using the one-pass theorem (no return theorem) developed by Nakanishi and Schlag $\href{ https://doi.org/10.1007/s00526-011-0424-9}{[38]}$.

Keywords

nonlinear Schrödinger equations, threshold solutions, global dynamics, ground state, blowup, scattering, special solutions, conservation laws, virial identity, one-pass theorem

2010 Mathematics Subject Classification

Primary 35B40, 35Q55. Secondary 35B44, 35P25.

Full Text (PDF format)

The authors would like to thank the anonymous referee for his/her careful reading. M. H. was supported by JSPS KEKENHI Grant Number JP22J00787. H.K. was supported by JSPS KAKENHI Grant Number JP20K03706. M.W was supported by JSPS KAKENHI Grant Number JP22J10027.

Received 25 March 2023

Published 1 November 2023