Global dynamics of a supercritical wave equation in a large data regime

TL;DR

This paper establishes the global existence of solutions to a supercritical nonlinear wave equation in three dimensions for large initial data, including dispersed and localized components, expanding understanding of wave dynamics in high-energy regimes.

Contribution

It proves global solutions exist for large initial data in the energy-supercritical regime, accommodating both dispersed and localized initial conditions.

Findings

Global solutions exist for large initial data in the supercritical regime.

Initial data can be decomposed into dispersed and localized parts.

Solutions are valid for data large in all homogeneous Sobolev norms .

Abstract

We prove the existence of global solutions to the nonlinear wave equation in $\mathbb{R}^{1+3}$ $$\Phi_{tt} - \Delta \Phi \pm \Phi|\Phi|^{p-1} = 0$$ in the energy-supercritical regime $p>5$, for a class of large initial data. Our initial data can be decomposed into two pieces, one which is dispersed in the sense of having large $L^2$ norm, while the other piece takes a localised short-pulse form. Consequently, we can obtain global existence for a class of initial data which is large in every homogeneous Sobolev norm $\dot{H}^s_x$ with $s \geq 0$.