Analysis Seminar

Consider the focusing semilinear wave equation in $R^3$ with energy-critical non-linearity $$\partial_t^2 \psi - \Delta \psi - \psi^5 = 0,\ \psi(0) = \psi_0,\ \partial_t \psi(0) = \psi_1.$$

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 $(\langle x\rangle^{-1}\dot{H}^1 \times \langle x\rangle^{-1} L^2)$. 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.