arXiv 1408.03312014-08math.AP

双曲空间上的能量临界半线性移位wave equation

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

Ruipeng Shen

备注：: 20 pages, 2 figures

摘要：

本文考虑双曲空间${\mathbb H}^n$上的半线性、能量临界、位移波动方程，其中$3 \leq n \leq 5$：\[ \partial_t^2 u - (Δ_{{\mathbb H}^n} + ρ^2) u = ζ|u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}. \]这里$ζ= \pm 1$和$ρ= (n-1)/2$是常数.我们在能量空间$H^{0,1} \times L^2 ({\mathbb H}^n)$中引入了一族与初始数据相容的Eschenhartz估计，然后用这些初始数据建立了一个局部理论。另外，在散焦情形$ζ= -1$中，我们证明了一个Morawetz型不等式\[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{ρ(\cosh |x|) |u(x,t)|^{2n/(n-2)}}{\sinh |x|} dμ(x) dt \leq n {\mathcal E}, \]，其中${\mathcal E}$是能量.此外，如果初始数据也是径向的，我们可以结合Morawetz型不等式，局部理论和径向$H^{0,1}({\mathbb H}^n)$函数的逐点估计来证明相应解的散射性.

英文摘要：

In this paper we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$: \[ \partial_t^2 u - (Δ_{{\mathbb H}^n} + ρ^2) u = ζ|u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}. \] Here $ζ= \pm 1$ and $ρ= (n-1)/2$ 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 \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{ρ(\cosh |x|) |u(x,t)|^{2n/(n-2)}}{\sinh |x|} dμ(x) dt \leq n {\mathcal E}, \] in the defocusing case $ζ= -1$, 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文