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

Abstract

Consider the focusing semilinear wave equation in R^3 with energy-critical nonlinearity \partial_t^2 ψ- Δψ- ψ^5 = 0, ψ(0) = ψ_0, \partial_t ψ(0) = ψ_1. This equation admits stationary solutions of the form ϕ(x, a) := (3a)^{1/4} (1+a|x|^2)^{-1/2}, called solitons, which solve the elliptic equation -Δϕ- ϕ^5 = 0. Restricting ourselves to the space of symmetric solutions ψfor which ψ(x) = ψ(-x), we find a local centre-stable manifold, in a neighborhood of ϕ(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.