A center-stable manifold for the energy-critical wave equation in ℝ3 in the symmetric setting

Abstract

Consider the focusing semilinear wave equation in ℝ3 with energy-critical nonlinearity [Formula: see text] This equation admits stationary solutions of the form [Formula: see text] called solitons, which solve the elliptic equation [Formula: see text] Restricting ourselves to the space of symmetric solutions ψ for which ψ(x) = ψ(-x), we find a local center-stable manifold, in a neighborhood of ϕ(x, 1), for this wave equation in the weighted Sobolev space [Formula: see text] Solutions with initial data on the manifold exist globally in time for t ≥ 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, recently introduced by Beceanu and Goldberg and adapted here to the case of Hamiltonians with a resonance.