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Summer 2016 Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three
Chenmin Sun, Bo Xia
Illinois J. Math. 60(2): 481-503 (Summer 2016). DOI: 10.1215/ijm/1499760018

Abstract

In this article, by following the strategies in dealing with supercritical cubic and quintic wave equations in (J. Eur. Math. Soc. (JEMS) 16 (2014) 1–30) and (J. Math. Pures Appl. (9) 105 (2016) 342–366), we obtain that, the equation \begin{equation*}(\partial^{2}_{t}-\Delta)u+|u|^{p-1}u=0,\quad3<p<5\end{equation*} is almost surely global well-posed with initial data $(u(0),\partial_{t}u(0))\in H^{s}(\mathbb{T}^{3})\times H^{s-1}(\mathbb{T}^{3})$ for any $s\in(\frac{p-3}{p-1},1)$. The key point here is that $\frac{p-3}{p-1}$ is much smaller than the critical index $\frac{3}{2}-\frac{2}{p-1}$ for $3<p<5$.

Citation

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Chenmin Sun. Bo Xia. "Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three." Illinois J. Math. 60 (2) 481 - 503, Summer 2016. https://doi.org/10.1215/ijm/1499760018

Information

Received: 3 August 2015; Revised: 1 March 2017; Published: Summer 2016
First available in Project Euclid: 11 July 2017

zbMATH: 06741646
MathSciNet: MR3680544
Digital Object Identifier: 10.1215/ijm/1499760018

Subjects:
Primary: 35L05, 42B37

Rights: Copyright © 2016 University of Illinois at Urbana-Champaign

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Vol.60 • No. 2 • Summer 2016
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