We consider the focusing energycritical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a oneparameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forwardintime solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st longtime random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.
more » « less NSFPAR ID:
 10123081
 Publisher / Repository:
 Oxford University Press
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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