A Semi-linear Shifted Wave Equation on the Hyperbolic Spaces with Application on a Quintic Wave Equation on \mathbb R²

2014/02/17 by Ruipeng Shen, Gigliola Staffilani, Shen, Ruipeng +1
Mathematics · #35L05 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1402.3879

openalex publication_date 2014/02/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space ∂_t² u - (Δ_\mathbb Hⁿ + ρ²) u = - |u|^p-1 u, (x,t)∈ \mathbb Hⁿ × \mathbb R; and introduce a Morawetz-type inequality ∫_-T-^T₊ ∫_\mathbb Hⁿ |u|^p+1 dμdt lt; C E, where E is the energy. Combining this inequality with a well-posedness theory, we can establish a scattering result for solutions with initial data in H^1/2,1/2 × H^1/2,-1/2(\mathbb Hⁿ) if 2 ≤ n ≤ 6 and 10. \]

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• Energy-critical semi-linear shifted wave equation on the hyperbolic spaces
• The wave equation on hyperbolic spaces
• Global existence for quasilinear wave equations on hyperbolic space
• Well-posedness and scattering for wave equations on hyperbolic spaces with singular data
• Existence and Scattering for Solutions to Semilinear Wave Equations on High Dimensional Hyperbolic Space