The Energy Critical Wave Equation in 3D

Abstract

This chapter discusses the energy critical nonlinear wave equation in 3 space dimensions. It mainly focuses on soliton resolution for radial solutions of nonlinear waves. For a long time there has been a widespread belief that global in time solutions of dispersive equations, asymptotically in time, decouple into a sum of finitely many modulated solitons, a free radiation term, and a term that goes to 0 at infinity. Such a result should hold for globally well-posed equations, or in general, with the additional condition that the solution does not blow-up. When blow-up may occur such decompositions are always expected to be unstable.