English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we consider a semi-linear, defocusing, shifted wave equation onthe 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  \[  \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} |u|^{p+1} d\mu dt < C E,  \] where $E$ is the energy. Combining this inequality with a well-posednesstheory, 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 thequintic wave equation $\partial_t^2 u - \Delta u = - |u|^4 u$ on ${\mathbbR}^2$ scatters if its initial data are radial and satisfy the conditions  \[  |\nabla u_0 (x)|, |u_1 (x)| \leq A(|x|+1)^{-3/2-\varepsilon};\quad |u_0 (x)|\leq A(|x|)^{-1/2-\varepsilon};\quad \varepsilon >0.  \]

A Semi-linear Shifted Wave Equation on the Hyperbolic Spaces with Application on a Quintic Wave Equation on ${\mathbb R}^2$