Center-stable manifold of the ground state in the energy space for the critical wave equation

TL;DR

This paper constructs a center-stable manifold for the ground state in the energy space of the critical wave equation, characterizing solutions near the ground state and serving as a threshold for scattering and blow-up behaviors.

Contribution

It develops a center-stable manifold without symmetry assumptions, extending previous classifications and including solutions with energies above the ground state.

Findings

Manifold contains all solutions scattering to ground states.

Includes solutions that blow up by concentration of ground states.

Extends to arbitrary energy levels with added radiation.

Abstract

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper. We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states.