# The focusing energy-critical nonlinear wave equation with random initial data

## TL;DR

This paper demonstrates that by randomizing initial data near solitons, one can with high probability obtain global solutions that scatter, extending the understanding of focusing nonlinear wave equations beyond small data scenarios.

## Contribution

It introduces a novel randomization method using distorted Fourier projections to establish long-time existence and scattering for focusing wave equations with large, randomized initial data.

## Key findings

- High probability of global solutions from randomized data near solitons
- First long-time random data existence result for focusing wave equations
- Solutions scatter after subtracting a modulated soliton

## Abstract

We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x(\mathbb{R}^3) \times H^{s-1}_x(\mathbb{R}^3)$, for any $s > 1/2$. By randomizing radial initial data in $ \dot H^s_x(\mathbb{R}^3) \times H^{s-1}_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 which give rise to global forward-in-time 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 first long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.