Global existence for solutions of the focusing wave equation with the compactness property

TL;DR

This paper proves that solutions with the compactness property for certain focusing wave and Schrödinger equations are globally existing, extending the understanding of solution behavior in critical and supercritical regimes.

Contribution

It establishes global existence results for solutions with the compactness property in focusing energy-critical wave and supercritical Schrödinger equations, a significant advancement in nonlinear PDE theory.

Findings

Solutions with the compactness property are global for the focusing energy-critical wave equation.

Similar global existence results are obtained for supercritical wave and Schrödinger equations.

The results contribute to the understanding of long-term behavior of solutions in critical regimes.

Abstract

We prove that every solution of the focusing energy-critical wave equation with the compactness property is global. We also give similar results for supercritical wave and Schr\"odinger equations.