The Strauss conjecture on negatively curved backgrounds
The Strauss conjecture on negatively curved backgrounds
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DOI：
10.3934/dcds.2019296
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发表时间：
2018-11
期刊：
影响因子：
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通讯作者：
Y. Sire;C. Sogge;Chengbo Wang
中科院分区：
文献类型：
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作者：
Y. Sire;C. Sogge;Chengbo Wang
本文主要讨论负曲率黎曼流形上半线性波动方程的几个小数据存在性结果。本文给出了双曲空间${\mathbb H}^n$上含有$\pm形式的非线性项的移动波动方程的任意幂$p\in（1，1+\frac{4}{n-1}]$的小数据整体存在性的一个简单的几何证明|u| ^p$或$\pm| u| ^{p-1}u$。它是基于Georgiev-Lindblad-Sogge（或Tataru）在欧氏空间上的加权Eschenhartz估计。我们还证明了一个小数据的存在定理，该定理扩展了Anker-Pierfelice和Metcalfe-Taylor的常曲率情况下的小数据存在定理。我们还讨论了曲率的作用，并提出了几个开放的问题。最后，在附录中，我们给出了Tataru对H ^3 $的色散估计的另一个证明，并解决了梅特卡夫-泰勒关于他的证明提出的一个对他有利的争议。我们的证明比Tataru的证明稍微更独立，因为它没有使用双曲空间上的重球面分析，如Harish-Chandra $c$-函数;相反，它只依赖于关于贝塞尔势的简单事实。
This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power $p\in (1, 1+\frac{4}{n-1}]$ for the shifted wave equation on hyperbolic space ${\mathbb H}^n$ involving nonlinearities of the form $\pm |u|^p$ or $\pm|u|^{p-1}u$. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge (or Tataru) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for the constant curvature case of Anker-Pierfelice and Metcalfe-Taylor. We also discuss the role of curvature and state a couple of open problems. Finally, in an appendix, we give an alternate proof of dispersive estimates of Tataru for ${\mathbb H}^3$ and settle a dispute, in his favor, raised in Metcalfe-Taylor about his proof. Our proof is slightly more self-contained than the one in Tataru since it does not make use of heavy spherical analysis on hyperbolic space such as the Harish-Chandra $c$-function; instead it relies only on simple facts about Bessel potentials.