Soliton resolution for the radial critical wave equation in all odd space dimensions
Thomas Duyckaerts, Carlos E. Kenig, Frank Merle
Abstract
Consider the energy-critical focusing wave equation in odd space dimension\n$N\\geq 3$. The equation has a nonzero radial stationary solution $W$, which is\nunique up to scaling and sign change. In this paper we prove that any radial,\nbounded in the energy norm solution of the equation behaves asymptotically as a\nsum of modulated $W$s, decoupled by the scaling, and a radiation term.\n The proof essentially boils down to the fact that the equation does not have\npurely nonradiative multisoliton solutions. The proof overcomes the fundamental\nobstruction for the extension of the 3D case (treated in our previous work,\nCambridge Journal of Mathematics 2013, arXiv:1204.0031) by reducing the study\nof a multisoliton solution to a finite dimensional system of ordinary\ndifferential equations on the modulation parameters. The key ingredient of the\nproof is to show that this system of equations creates some radiation,\ncontradicting the existence of pure multisolitons.\n