Profile decompositions for wave equations on hyperbolic space with applications

TL;DR

This paper develops profile decompositions for wave equations on hyperbolic space, including perturbed cases, and uses them to prove global well-posedness and scattering for energy-critical nonlinear waves in three dimensions.

Contribution

It introduces a general framework for profile decompositions on hyperbolic space and applies it to establish scattering results for energy-critical nonlinear wave equations.

Findings

Proved profile decompositions for waves on hyperbolic space.

Established global well-posedness and scattering for the energy-critical wave equation.

Extended results to perturbed hyperbolic spaces with potentials.

Abstract

The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential. Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive time-independent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves in infinite time. This proof will serve as a blueprint for the arguments in a forthcoming work, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.