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-down in finite time.
The first author was supported in part by NSF and the second one in part by CNRS and by ANR ONDENONLIN. Part of this research was carried out during visits of the second author to the University of Chicago and I.H.E.S. and of the first author to Paris XIII.
"Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation." Acta Math. 201 (2) 147 - 212, 2008. https://doi.org/10.1007/s11511-008-0031-6