Profile decompositions for wave equations on hyperbolic space with applications

Abstract

Abstract. The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gérard 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 timeindependent 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 → ±∞. 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.

Equivariant wave maps on the hyperbolic plane with large energy

“…A key ingredient in the proof of Proposition 3.2 is a linear profile decomposition for (3.3), which was first proved in the case of the wave equation on R × R 3 by Bahouri and Gérard [2], and the Schrödinger equation on R× H 3 by Ionescu, Pausader and Staffilani [7]. In [12,Theorem 4.2], we proved a linear profile decomposition for a fairly general class of wave equations on R × H d , including wave equations with spectral shifts and perturbations by a time-independent potential. Here we only state a simpler version that suffices for our use.…”

“…Our proof of Proposition 3.2 relies on the concentration compactness method [8,9]. Execution of this strategy in this setting requires several ingredients, which were mostly established in the previous work [12,13] of the authors.…”

“…If the theorem fails, then by Proposition 3.2 we can find a critical element u * such that u * ≡ (0, 0) and the forward trajectory K + is pre-compact in H 1 × L 2 . Let ψ * = sinh r u * + (P λ , 0) be the corresponding equivariant wave map with energy E. We will argue as in the proof of Proposition 8.4 in [12] to show that u * (t) ≡ (0, 0), which is a contradiction. All the constants in the proof may depend on the energy E of the wave map ψ * .…”

A semi-linear energy critical wave equation with an application

“…The point is that dilations and space translations are no longer contained in the symmetric group of this equation. The situation is similar when people are considering the compactness process for wave/Schrödinger equations on a space other than the Euclidean spaces, see [27,42], for instance. We start by introducing the profile decomposition, before more details are discussed.…”

Conditional Stable Soliton Resolution for a Semi-linear Skyrme Equation

“…This gives rise to two different types of profiles at the level of the linear evolution (in [30] there are two different types of profiles only for the nonlinear profile decomposition; see [30,Section 4.1.2]). A similar situation where multiple types of linear profiles arise has been treated in [31] in the setting of non-radial waves with potential on hyperbolic space, and we refer the reader to the arguments in [31,Section 4.1] for how to include a potential V into the linear profile decomposition. A straightforward implementation of arguments from [31] (to deal with the potential) into the detailed scheme in [30] covers the proof of Proposition 5.1.…”