Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

Kenji  Nakanishi; Tristan  Roy

doi:10.3934/cpaa.2016026

Article Contents

2016, Volume 15, Issue 6: 2023-2058. Doi: 10.3934/cpaa.2016026

This issue Previous Article Uniform global existence and convergence of Euler-Maxwell systems with small parameters Next Article On the Hardy-Littlewood-Sobolev type systems

Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

Kenji Nakanishi^1, and
Tristan Roy^2,

1.
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Osaka University, Suita, Osaka 565-0871
2.
Graduate School of Mathematics, Nagoya University

Received: September 30, 2015

Revised: May 31, 2016

Published: October 2016

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Abstract

Consider the focusing energy critical Schrödinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with [19], the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference between [19] and this paper is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.

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Mathematics Subject Classification: Primary: 35Q55.

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