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- W2023265854 abstract "In this paper we consider a semi-linear, defocusing, shifted wave equation on the hyperbolic space [ partial_t^2 u - (Delta_{{mathbb H}^n} + rho^2) u = - |u|^{p-1} u, quad (x,t)in {mathbb H}^n times {mathbb R}; ] and introduce a Morawetz-type inequality [ int_{-T_-}^{T_+} int_{{mathbb H}^n} |u|^{p+1} dmu dt < 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 $H^{1/2,1/2} times H^{1/2,-1/2}({mathbb H}^n)$ if $2 leq n leq 6$ and $1<p<p_c = 1+ 4/(n-2)$. As another application we show that a solution to the quintic wave equation $partial_t^2 u - Delta u = - |u|^4 u$ on ${mathbb R}^2$ scatters if its initial data are radial and satisfy the conditions [ |nabla u_0 (x)|, |u_1 (x)| leq A(|x|+1)^{-3/2-varepsilon};quad |u_0 (x)| leq A(|x|)^{-1/2-varepsilon};quad varepsilon >0. ]" @default.
- W2023265854 created "2016-06-24" @default.
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- W2023265854 date "2014-02-16" @default.
- W2023265854 modified "2023-10-16" @default.
- W2023265854 title "A Semi-linear Shifted Wave Equation on the Hyperbolic Spaces with Application on a Quintic Wave Equation on ${mathbb R}^2$" @default.
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- W2023265854 doi "https://doi.org/10.48550/arxiv.1402.3879" @default.
- W2023265854 hasPublicationYear "2014" @default.
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