Energy-critical semi-linear shifted wave equation on the hyperbolic spaces

الأصل · EN

In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space Hⁿ with 3 ≤ n ≤ 5: ∂ₜ² u - (ΔHⁿ + ρ²) u = ζ|u|⁴/⁽ⁿ⁻²⁾ u, (x,t)∈ Hⁿ × R. Here ζ= ± 1 and ρ= (n-1)/2 are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space H⁰,¹ × L² (Hⁿ) and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality ∫₋ₜ₋ᵗ⁺ ∫Hⁿ ρ(|x|) |u(x,t)|²ⁿ/⁽ⁿ⁻²⁾ |x| dμ(x) dt ≤ n E, in the defocusing case ζ= -1, where E is the energy. Moreover, if the initial data are also radial, we can prove the scattering of the corresponding solutions by combining the Morawetz-type inequality, the local theory and a pointwise estimate on radial H⁰,¹(Hⁿ) functions.