Large Outgoing Solutions to Supercritical Wave Equations

TL;DR

This paper establishes the existence of global solutions for the energy-supercritical wave equation in three spatial dimensions, accommodating large and even infinite initial data norms by characterizing outgoing solutions.

Contribution

It introduces a new class of outgoing initial data and proves global existence for supercritical wave equations with large or infinite critical norms.

Findings

Global solutions exist for a broad class of outgoing initial data.

Solutions can have arbitrarily large critical Sobolev, Besov, Lebesgue, and Lorentz norms.

The class of initial data includes those with infinite critical norms.

Abstract

We prove the existence of global solutions to the energy-supercritical wave equation in R^{3+1} u_{tt}-\Delta u + |u|^N u = 0, u(0) = u_0, u_t(0) = u_1, 4<N<\infty, for a large class of radially symmetric finite-energy initial data. Functions in this class are characterized as being outgoing under the linear flow --- for a specific meaning of "outgoing" defined below. In particular, we construct global solutions for initial data with large (even infinite) critical Sobolev, Besov, Lebesgue, and Lorentz norms and several other large critical norms.