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Full range of blow up exponents for the quintic wave equation in three dimensions

For the critical focusing wave equation \Box u = u^5 on R^{3+1} in the radial case, we prove the existence of type II blow up solutions with scaling parameter λ(t) = t^{-1-ν} for all ν>0. This extends the previous work by the authors and Tataru where the condition ν> 1/2 had been imposed, and gives the optimal range of polynomial blow up rates in light of recent work by Duyckaerts, Kenig and Merle.