Soliton Resolution for the Radial Quadratic Wave Equation in Space Dimension 6
Résumé
We consider the quadratic semilinear wave equation in six dimensions. This energy critical problem admits a ground state solution, which is the unique (up to scaling) positive stationary solution. We prove that any spherically symmetric solution, that remains bounded in the energy norm, evolves asymptotically to a sum of decoupled modulated ground states, plus a radiation term. As a by-product of the approach we prove the non-existence of multisoliton solutions that do not emit any radiation. The proof follows the method initiated for large odd dimensions by the last three authors, reducing the problem to ruling out the existence of such non-radiative multisolitons, by deriving a contradiction from a finite dimensional system of ordinary differential equations governing their modulation parameters. In comparison, the difficulty in six dimensions is the failure of certain channel of energy estimates and the related existence of a linear resonance. We use the obtention of new channel of energy estimates, from our previous article [CDKM22a], as well as the classification of non-radiative solutions with small energy, from our work [CDKM22b]. Contents C. COLLOT, T. DUYCKAERTS, C. KENIG, AND F. MERLE 3.3. Non-radiative solutions 17 4. Proof of the soliton resolution 23 4.1. Setting of the proof 24 4.2. Expansion along a sequence of times and renormalisation 25 4.3. Estimates on λ j and β j 27 4.4. System of equations and estimates on the derivatives 31 4.5. Restriction on the set of indices and end of the proof 35 4.6. Proof of the soliton resolution for the |u|u nonlinearity 37 4.7. Proof of the rigidity result for global non-radiative solutions 39 Appendix A. Study of a system of differential inequalities 39 Appendix B. A few estimates 42 References 43 4 N-2 u in general space dimension N ≥ 3, whose ground state is given by W (x) = 1 + |x| 2 N (N -2) 1-N 2 .
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