Profiles of bounded radial solutions of the focusing, energy-critical wave equation

Abstract

Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W , then the decomposition holds without a restriction to a subsequence.

Stable soliton resolution for equivariant wave maps exterior to a ball

“…However, such dynamical identities are extremely sensitive to the precise structure of the particular nonlinear wave equation under consideration, and do not easily generalize more complicated nonlinearities. An important breakthrough was made by Duyckaerts, Kenig, and Merle, [4,6,5,7,8], who developed an alternative approach called the 'channels of energy' method for rigidity arguments.…”

“…Consequently, (1.1) gives the following estimate from [4,Lemma 4.2], see also [5,7,8]: for any a ≥ 0 one has…”

Corrigendum: On the Soliton Resolution for Equivariant Wave Maps to the Sphere

“…1 The failure of the above Pythagorean expansions (2.1)-(2.2) created a gap in the proofs of [5], which the corrected version [6] filled. That failure also has an impact in the context of equivariant wave maps [2][3][4], which [6] also corrects, mutatis mutandis. For the convenience of the reader, we provide here an independent, shorter correction that applies to [2].…”

“…Indeed, in both [5] and [2], the proof consists in first constructing a nonlinear decomposition of a solution, with an error controlled in a "weak space," and in a second step, to prove that the error actually tends to 0 in the energy space. The failure of (2.1)-(2.2) impacts the second step.…”

“…The failure of (2.1)-(2.2) impacts the second step. But an important difference is that the "weak" space in [5] is a Strichartz space, whereas in [2] it is known after the first step that the error has no energy at all scales [2, theorem 3.3], which is a much stronger topology. Therefore the gap in [2] is smaller and can be filled in a faster way.…”