A semi-linear shifted wave equation on the hyperbolic spaces with application on a quintic wave equation on R²

Abstract

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space ∂[subscript t][superscript 2]u- (∆ℍ[superscript n] +ρ[superscript 2] )u = -|u| p[superscript -1] u, (x,t) ∈ ℍ n × ℝ, and we introduce a Morawetz-type inequality (Formula presented) where ε is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in H[superscript 1/2,1/2] × H[superscript 1/2,−1/2](ℍ[superscript n) if 2 ≤ n ≤ 6 and 1 < p < p[subscript c] = 1+4/(n − 2). As another application we show that a solution to the quintic wave equation ∂[subscript t][superscript 2]u − Δu = −|u|[superscript 4] u on ℝ[superscript 2] scatters if its initial data are radial and satisfy the conditions |∇u[subscript 0](x)|, |u[subscript 1](x)| ≤ A(|x| + 1) [superscript −3/2−ε] , |u[subscript 0](x)| ≤ A(|x|)[superscript −1/2−ε], ε > 0.