A remark on barely Ḣsp-supercritical wave equations

Roy, Tristan

doi:10.2140/apde.2012.5.199

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A remark on barely $\dot H^{s_{p}}$-supercritical wave equations

Tristan Roy

Vol. 5 (2012), No. 1, 199–218

DOI: 10.2140/apde.2012.5.199

Abstract

We prove that a good $Ḣ^{s_{p}}$ critical theory for the $3 D$ wave equation $\partial_{t t} u - △ u = - | u |^{p - 1} u$ can be extended to prove global well-posedness of smooth solutions of at least one $3 D$ barely $Ḣ^{s_{p}}$ -supercritical wave equation $\partial_{t t} u - △ u = - | u |^{p - 1} u g (| u |)$ , with $g$ growing slowly to infinity, provided that a Kenig-Merle type condition is satisfied. This result is related to those obtained by Tao and the author for the particular case $s_{p} = 1$ , showing global regularity for $g$ growing logarithmically with radial data and for $g$ growing doubly logarithmically with general data.

Keywords

wave equation, global existence, barely supercritical

Mathematical Subject Classification 2000

Primary: 35Q55

Milestones

Received: 26 April 2010

Revised: 17 July 2010

Accepted: 16 August 2010

Published: 25 June 2012

Proposed: Terence Tao

Authors

Tristan Roy
School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ 08540 Institute For Advanced Study

Vol. 5, No. 1, 2012