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Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations

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  • Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.
    Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 35L71, 35Q55, 35B44.

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