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Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

Carlos E. Kenig Department of Mathematics, University of Chicago Frank Merle Département de Mathématiques, Université de Cergy-Pontoise

TBD mathscidoc:1701.331998

Acta Mathematica, 201, (2), 147-212, 2006.10
We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution$W$which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space$H$^{1}than the one of$W$, then we have global well-posedness and scattering. If the norm is larger than the one of$W$, then we have break-down in finite time.
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@inproceedings{carlos2006global,
  title={Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation},
  author={Carlos E. Kenig, and Frank Merle},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203352468039707},
  booktitle={Acta Mathematica},
  volume={201},
  number={2},
  pages={147-212},
  year={2006},
}
Carlos E. Kenig, and Frank Merle. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. 2006. Vol. 201. In Acta Mathematica. pp.147-212. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203352468039707.
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