Scattering and blowup problems for a class of nonlinear Schrödinger equations.
type: Item
instance of: scholarly article
title: Scattering and blowup problems for a class of nonlinear Schrödinger equations.
zbMATH Open document ID: 1274.35353
author: T. Akahori
author: H. Kikuchi
author: H. Nawa
published in: Differential and Integral Equations
publication date: 2013-03-21T00:00:00Z
review text: We study the scattering and blow-up problem for the equation \[ 2\text{i}\psi _t(x,t)+\Delta \psi (x,t)+f(\psi (x,t))=0,\quad x\in \mathbb {R}^d,\quad t\in \mathbb {R}, \] where \(f\: \mathbb {C}\rightarrow \mathbb {C}\) is a continuously differentiable function more general than pure power nonlinearity. We prove the existence of a standing-wave solution of minimal action (ground state) for any frequency \(\omega >0\). We define two invariant regions \(A_{\omega ,+}\) and \(A_{\omega ,-}\) for each \(\omega > 0\) in \(H^1(\mathbb {R}^d)\) such that any solution starting from \(A_{\omega ,+}\) behaves asymptotically free as \(t\rightarrow \pm \infty \), one from \(A_{\omega ,-}\) blows up or grows up, and the ground state belongs to \(\overline {A_{\omega ,+}}\bigcap \overline {A_{\omega ,-}}\).
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 6148068
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: ground state
zbMATH Keywords: variational estimate
zbMATH Keywords: minimizer
zbMATH Keywords: invariant set
zbMATH Keywords: blow-up
reviewed by: Marie Kopáčková
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6148068