Profile decompositions for wave equations on hyperbolic space with applications

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 as $$t \rightarrow \pm \infty $$ . This proof serves as a blueprint for the arguments in the work [45] by the authors, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.