Wave and Klein–Gordon Equations on Certain Locally Symmetric Spaces

Abstract

This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global wellposedness results for the corresponding semilinear equation with low regularity data as on real hyperbolic spaces.2000 Mathematics Subject Classification. 35Q55, 43A85, 22E30, 35P25, 47J35, 58D25.

Wave equation on certain noncompact symmetric spaces

“…Consider the Poincaré series P (s; x, y) = γ∈Γ e −sd(x,γy) , s > 0, x, y ∈ X, and denote by δ(Γ) = inf{s > 0 | P (s; x, y) < +∞} its critical exponent. In [Zha20], the author has studied the wave equation and has obtained similar Strichartz inequality and global well-posedness results as in Sect. 5, in the case where Γ\X is a rank one locally symmetric space such that Γ is convex cocompact and δ(Γ) < |ρ|.…”

“…Notice that we could deduce the last condition (3) from the convex cocompactness of Γ in rank one. We refer to [Zha20] and the references therein for more details about wave type equations on locally symmetric spaces.…”

Wave equation on certain noncompact symmetric spaces

“…Consider the Poincaré series P (s; x, y) = γ∈Γ e −sd(x,γy) , s > 0, x, y ∈ X, and denote by δ(Γ) = inf{s > 0 | P (s; x, y) < +∞} its critical exponent. In [Zha19], the author has studied the wave equation and has obtained similar Strichartz inequality and global well-posedness results as in Sect. 5, in the case where Γ\X is a rank one locally symmetric space such that Γ is convex cocompact and δ(Γ) < |ρ|.…”

“…However, thanks to our wave kernel estimates Theorem 3.1 and Theorem 3.3, we can study along the lines of [Zha19] the wave equation on higher rank noncompact locally symmetric spaces, under slightly different assumptions:…”

“…Notice that we could deduce the last condition (3) from the convex cocompactness of Γ in rank one. We refer to [Zha19] and the references therein for more details about wave type equations on locally symmetric spaces.…”