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

TL;DR

This paper investigates the speed at which type II blow-up solutions occur for the energy-critical wave equation in radial symmetry, providing optimal bounds in five dimensions and constraints on the asymptotic profile.

Contribution

It establishes bounds on blow-up speed for radial solutions with regular profiles and shows such profiles cannot be strictly negative at the origin, advancing understanding of blow-up dynamics.

Findings

Bounds on blow-up speed are proven, optimal in 5D.

Regularity of the profile constrains its sign at the origin.

Results apply to radial solutions in dimensions 3, 4, 5.

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.