## Differential and Integral Equations

### Energy-critical semi-linear shifted wave equation on the hyperbolic spaces

Ruipeng Shen

#### Abstract

In this paper, we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$: $\partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = \zeta |u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}.$ Here, $\zeta = \pm 1$ and $\rho = (n-1)/2$ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space $H^{0,1} \times L^2 ({\mathbb H}^n)$ and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality $\int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{\rho (\cosh |x|) |u(x,t)|^{2n/(n-2)}} {\sinh |x|} d\mu(x) dt \leq n {\mathcal E},$ in the defocusing case $\zeta = -1$, 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 $H^{0,1}({\mathbb H}^n)$ functions.

#### Article information

Source
Differential Integral Equations Volume 29, Number 7/8 (2016), 731-756.

Dates
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.die/1462298683

Mathematical Reviews number (MathSciNet)
MR3498875

Zentralblatt MATH identifier
06604493

#### Citation

Shen, Ruipeng. Energy-critical semi-linear shifted wave equation on the hyperbolic spaces. Differential Integral Equations 29 (2016), no. 7/8, 731--756. https://projecteuclid.org/euclid.die/1462298683.