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

Authors
  • Roy, Tristan
Type
Preprint
Publication Date
Sep 04, 2009
Submission Date
Oct 28, 2008
Source
arXiv
License
Yellow
External links

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.

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