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

@article{Kenig2006GlobalWS,
title={Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation},
author={Carlos E. Kenig and Frank Merle},
journal={Acta Mathematica},
year={2006},
volume={201},
pages={147-212}
}
• Published 26 October 2006
• Mathematics
• Acta Mathematica
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 H1 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…
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