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

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.

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
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