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.
Threshold solutions for the nonlinear Schrödinger equation
We study the focusing NLS equation in $R\mathbb{R}^N$ in the masssupercritical and energysubcritical (or intercritical ) regime, with $H^1$ data at the massenergy threshold $\mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation.
In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $Q^\pm$, besides the standing wave $e^{it}Q$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blowup occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases.
These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixedpoint argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations.
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 Award ID(s):
 1927258
 NSFPAR ID:
 10339544
 Date Published:
 Journal Name:
 Revista Matemática Iberoamericana
 ISSN:
 02132230
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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