Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space
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| Format: | Preprint |
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2024
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| author | Jia, Qiuye Zhang, Junyong |
| author_facet | Jia, Qiuye Zhang, Junyong |
| contents | We study the pointwise decay estimates for the Schrödinger and wave equations on a product cone $(X,g)$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. Under the assumption that the conjugate radius $ε$ of $Y$ satisfies $ε>π$, we prove the pointwise dispersive estimates for the Schrödinger and half-wave propagator in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come to play if $ε>π$. In a work in progress, we will further study the case that $ε\leqπ$ in which the role of conjugate points come. A new finding is that a threshold of the conjugate radius of $Y$ for $L^p$-estimates in this setting is the magical number $π$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space Jia, Qiuye Zhang, Junyong Analysis of PDEs Differential Geometry Spectral Theory 35B25, 35K08, 35Q41, 35K20, 33C10, 35S30, 35L05 We study the pointwise decay estimates for the Schrödinger and wave equations on a product cone $(X,g)$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. Under the assumption that the conjugate radius $ε$ of $Y$ satisfies $ε>π$, we prove the pointwise dispersive estimates for the Schrödinger and half-wave propagator in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come to play if $ε>π$. In a work in progress, we will further study the case that $ε\leqπ$ in which the role of conjugate points come. A new finding is that a threshold of the conjugate radius of $Y$ for $L^p$-estimates in this setting is the magical number $π$. |
| title | Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space |
| topic | Analysis of PDEs Differential Geometry Spectral Theory 35B25, 35K08, 35Q41, 35K20, 33C10, 35S30, 35L05 |
| url | https://arxiv.org/abs/2411.16029 |