Nondispersive decay for the cubic wave equation

TL;DR

This paper studies the cubic focusing wave equation's behavior with hyperboloidal initial data, proving the existence of a special manifold of initial conditions that produce global solutions which do not disperse as free waves.

Contribution

It establishes the existence of a co-dimension 4 Lipschitz manifold of initial data leading to non-scattering global solutions without symmetry assumptions.

Findings

Existence of a co-dimension 4 Lipschitz manifold of initial data.

Global solutions that do not scatter to free waves.

Results hold without symmetry assumptions.

Abstract

We consider the hyperboloidal initial value problem for the cubic focusing wave equation. Without symmetry assumptions, we prove the existence of a co-dimension 4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves.