A Centre-Stable Manifold for the Energy-Critical Wave Equation in R^3 in the Symmetric Setting

TL;DR

This paper constructs a local center-stable manifold near a soliton solution for the energy-critical wave equation in R^3 under symmetric conditions, demonstrating global existence and stability of solutions on this manifold.

Contribution

It introduces a new class of reverse Strichartz estimates for Hamiltonians with resonances, enabling the construction of a center-stable manifold in the symmetric setting.

Findings

Solutions on the manifold exist globally for t ≥ 0

Solutions depend continuously on initial data

Solutions can be decomposed into a rescaled soliton and dispersive radiation

Abstract

Consider the focusing semilinear wave equation in R^3 with energy-critical nonlinearity \partial_t^2 \psi - \Delta \psi - \psi^5 = 0, \psi(0) = \psi_0, \partial_t \psi(0) = \psi_1. This equation admits stationary solutions of the form \phi(x, a) := (3a)^{1/4} (1+a|x|^2)^{-1/2}, called solitons, which solve the elliptic equation -\Delta \phi - \phi^5 = 0. Restricting ourselves to the space of symmetric solutions \psi for which \psi(x) = \psi(-x), we find a local centre-stable manifold, in a neighborhood of \phi(x, 1), for this wave equation in the weighted Sobolev space <x>^{-1} \dot H^1 \times <x>^{-1} L^2. Solutions with initial data on the manifold exist globally in time for t \geq 0, depend continuously on initial data, preserve energy, and can be written as the sum of a rescaled soliton and a dispersive radiation term. The proof is based on a new class of reverse Strichartz estimates, introduced in Beceanu-Goldberg and adapted here to the case of Hamiltonians with a resonance.