Soliton resolution for the energy-critical nonlinear wave equation in the radial case

TL;DR

This paper proves that solutions to the energy-critical nonlinear wave equation in higher dimensions decompose into ground states and free radiation over time, confirming the soliton resolution conjecture in the radial case.

Contribution

It establishes the soliton resolution for the energy-critical nonlinear wave equation with radial symmetry in dimensions four and higher, a significant step in understanding long-term dynamics.

Findings

Solutions decompose into ground states and radiation asymptotically

Validates soliton resolution conjecture in the radial case

Applicable for space dimensions D ≥ 4

Abstract

We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions $D \ge 4$. This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution $W$, called the ground state. We prove that every finite energy solution with bounded energy norm resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.