Global Existence Of Smooth Solutions Of A 3D Loglog Energy-Supercritical Wave Equation

TL;DR

This paper proves the global existence of smooth solutions for a 3D loglog energy-supercritical wave equation with specific initial data, using long-time estimates and induction on time of Strichartz estimates.

Contribution

It introduces a method to control the solution's norms over time, establishing global existence for a class of supercritical wave equations.

Findings

Global existence of smooth solutions is proven.

A new approach combines potential decay estimates with induction on time.

The method applies to wave equations with specific supercritical nonlinearities.

Abstract

We prove global existence of smooth solutions of the 3D loglog energy-supercritical wave equation $\partial_{tt} u - \triangle u = -u^{5} \log^{c} (log(10+u^{2})) $ with $0 < c < {8/225}$ and smooth initial data $(u(0)=u_{0}, \partial_{t} u(0)=u_{1})$. First we control the $L_{t}^{4} L_{x}^{12}$ norm of the solution on an arbitrary size time interval by an expression depending on the energy and an \textit{a priori} upper bound of its $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm, with $\tilde{H}^{2}(\mathbb{R}^{3}):=\dot{H}^{2}(\mathbb{R}^{3}) \cap \dot{H}^{1}(\mathbb{R}^{3})$. The proof of this long time estimate relies upon the use of some potential decay estimates \cite{bahger, shatstruwe} and a modification of an argument in \cite{taolog}. Then we find an \textit{a posteriori} upper bound of the $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.