Multi-bubble solutions are constructed for the 4D energy-critical wave equation blowing up at N symmetric points with log(1/λ(t)) = (9c/4)^{1/3} t^{2/3} + O(t^{1/3}).
Rigidity of the multi-bubble solutions to the energy critical wave equation in dimension five
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abstract
We study the asymptotic dynamics of multi-bubble solutions to the focusing energy-critical wave equation in five dimensions. Assuming that the solution asymptotically decomposes into a finite superposition of spatially separated bubbles with comparable scales, we prove a rigidity result that describes the precise long-time behavior of these scales. More precisely, we show that all scaling parameters are necessarily of order $t^{-2}$, and that the corresponding renormalized modulation vector converges to a connected component of a finite-dimensional algebraic set determined by the limiting spatial configuration of the bubbles. This algebraic system encodes the strong interactions between the polynomial tails of the bubbles and governs the effective asymptotic dynamics of the multi-bubble regime.
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math.AP 1years
2026 1verdicts
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Construction of multi-bubble solutions for the energy-critical wave equation in dimension four
Multi-bubble solutions are constructed for the 4D energy-critical wave equation blowing up at N symmetric points with log(1/λ(t)) = (9c/4)^{1/3} t^{2/3} + O(t^{1/3}).