Global existence of solutions of a loglog energy-supercritical Klein-Gordon equation

TL;DR

This paper proves the global existence of solutions for a loglog energy-supercritical Klein-Gordon equation in dimensions 3 to 5 by analyzing blow-up behavior and establishing bounds on a Strichartz-type norm.

Contribution

It introduces a novel analysis near blow-up time to establish global solutions for a challenging class of Klein-Gordon equations.

Findings

Global existence proven for n=3,4,5

Bounded Strichartz norm near blow-up time

Contradiction established assuming finite-time blow-up

Abstract

We prove global existence of solutions of a loglog energy-supercritical Klein-Gordon equation for n=3,4,5. Assuming that blow-up occurs at a time of maximal existence, we perform an analysis close to this time in order to find a finite bound of a Strichartz-type norm, which eventually leads to a contraction with the blow-up assumption.