A CentreStable Manifold for the EnergyCritical Wave Equation in R^3 in the Symmetric Setting
Abstract
Consider the focusing semilinear wave equation in R^3 with energycritical 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+ax^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 centrestable 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 BeceanuGoldberg and adapted here to the case of Hamiltonians with a resonance.
 Publication:

arXiv eprints
 Pub Date:
 December 2012
 DOI:
 10.48550/arXiv.1212.2285
 arXiv:
 arXiv:1212.2285
 Bibcode:
 2012arXiv1212.2285B
 Keywords:

 Mathematics  Analysis of PDEs;
 35L05;
 35C08;
 37K40