Global existence of solutions of a loglog energy-supercritical Klein-Gordon equation
Tristan Roy
TL;DR
The paper tackles global existence for the defocusing loglog energy-supercritical Klein-Gordon equation in dimensions $n\in\{3,4,5\}$ with a barely supercritical nonlinearity $-|u|^{\frac{4}{n-2}}u\,g(|u|)$ where $g(|u|)=\log^{\gamma}(\log(10+|u|^{2}))$ and $0<\gamma<\gamma_n$. It develops a framework that combines Jensen-type inequalities and fractional Leibniz rules with Strichartz estimates to bound a Strichartz-type norm on short time intervals, then patches these estimates via an iteration to global time using a concentration/Morawetz-type mechanism near a hypothetical blow-up. A critical ingredient is showing that, under a priori bounds, the long-time Strichartz norm remains finite, which precludes blow-up and yields global well-posedness for $0<\gamma<\gamma_n$ and data in $H^{k}\times H^{k-1}$ with $k$ in an appropriate range $I_n$. The results extend the understanding of barely energy-supercritical dispersive dynamics by providing a contraction-based route to global existence in the KG setting and by linking short-time analysis to global behavior through a delicate combination of frequency localization, Jensen-type estimates, and Morawetz-type controls.
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.