2014/08/02 by Ruipeng Shen, Shen, Ruipeng
Mathematics · #Advanced Mathematical Physics Problems
paper · pdf · doi:10.48550/arxiv.1408.0331
In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space \mathbb Hn with 3 ≤ n ≤ 5: ∂t2 u - (Δ_\mathbb Hn + ρ2) u = ζ|u|4/(n-2) u, (x,t)∈ \mathbb Hn × \mathbb 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 (\mathbb Hn) and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality ∫-T-T+ ∫_\mathbb Hn \fracρ(\cosh |x|) |u(x,t)|2n/(n-2)\sinh |x| dμ(x) dt ≤ n \mathcal E, in the defocusing case ζ= -1, where \mathcal 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(\mathbb Hn) functions.