Profiles for the radial focusing energy-critical wave equation in odd dimensions. (English) Zbl 1348.35033

Rodriguez, Casey

Adv. Differ. Equ. 21, No. 5-6, 505-570 (2016).

Considering the radial solutions of the focusing energy critical wave equation \(u_{tt}-\Delta u=|u|^{4/(N-2)}u\) posed for \(x\in \mathbb{R}^N\) with odd \(N\geq 5\). In this paper, the author proves that if the solution remains bounded in the energy space as we approach the maximal forward time of existence (which is known as type-II solutions), then along a sequence of times converging to the maximal forward time of existence, the solution decouples 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 a bound on the evolution that rules out the formation of multiple solitons, then this decoupling holds for all times approaching the maximal forward time of existence.

Reviewer: Chengbo Wang (Hangzhou)

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35L71	Second-order semilinear hyperbolic equations
35L15	Initial value problems for second-order hyperbolic equations
35B07	Axially symmetric solutions to PDEs

Keywords:

profile decomposition; type II solutions; dynamically rescaled solitons