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

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle

Published 2013-11-04, updated 2014-03-21Version 3

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 Schr\"odinger equation.