Bounds on the speed of type II blow-up for the energy critical wave equation in the radial case

Abstract: We consider the focusing energy-critical wave equation in space dimension $N\in {3, 4, 5}$ for radial data. We study type II blow-up solutions which concentrate one bubble of energy. It is known that such solutions decompose in the energy space as a sum of the bubble and an asymptotic profile. We prove bounds on the blow-up speed in the case when the asymptotic profile is sufficiently regular. These bounds are optimal in dimension $N = 5$. We also prove that if the asymptotic profile is sufficiently regular, then it cannot be strictly negative at the origin.