Lectures on the Energy Critical Nonlinear Wave Equation

Examines recent advances in the study of the long-time asymptotics of large solutions to critical nonlinear dispersive equations. The first part of the monograph describes, in the context of the energy critical wave equation, the “concentration-compactness/rigidity theorem method”. The second part of the monograph describes the ``channel of energy'' method.

This monograph deals with recent advances in the study of the long-time asymptotics of large solutions to critical nonlinear dispersive equations. The first part of the monograph describes, in the context of the energy critical wave equation, the ``concentration-compactness/rigidity theorem method'''' introduced by C. Kenig and F. Merle. This approach has become the canonical method for the study of the ``global regularity and well-posedness'''' conjecture (defocusing case) and the ``ground-state'''' conjecture (focusing case) in critical dispersive problems.

The second part of the monograph describes the ``channel of energy'''' method, introduced by T. Duyckaerts, C. Kenig, and F. Merle, to study soliton resolution for nonlinear wave equations. This culminates in a presentation of the proof of the soliton resolution conjecture, for the three-dimensional radial focusing energy critical wave equation.

It is the intent that the results described in this book will be a model for what to strive for in the study of other nonlinear dispersive equations.