A center-stable manifold for the energy-critical wave equation in R-3 in the symmetric setting

作者：Beceanu Marius^*

来源：Journal of Hyperbolic Differential Equations, 2014, 11(3): 437-476.

DOI：10.1142/S021989161450012X

• Center-stable manifold
• wave equation
• semi-linear wave equation
• energy-critical wave equation

Consider the focusing semilinear wave equation in R-3 with energy-critical nonlinearity
partial derivative(2)(t)psi - Delta psi - psi(5) = 0, psi(0) = psi(0), partial derivative(t) psi(0) = psi(1).
This equation admits stationary solutions of the form
phi(x, a) := (3a)(1/4) (1 + a vertical bar x vertical bar(2))(-1/2),
called solitons, which solve the elliptic equation
-Delta phi - phi(5) = 0.
Restrictin, ourselves to the space of symmetric solutions psi for which psi(x) = psi(-x) , we find a local center-stable manifold, in a neighborhood of phi(x, 1), for this wave equation in the weighted Sobolev space < x >(-1) (H) over dot(1) x < x >(-1) L-2. 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

• 出版日期2014-9