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

Abstract

Consider the focusing energy critical Schrödinger equation in three space dimensionswith 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 allowsto 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 describethe 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.

Locations

• Communications on Pure &amp Applied Analysis - View - PDF
• arXiv (Cornell University) - View - PDF