Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914080701284352 |
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| author | Casteras, Jean-Baptiste Holopainen, Ilkka |
| author_facet | Casteras, Jean-Baptiste Holopainen, Ilkka |
| contents | We study the fourth order Schrödinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_13804 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds Casteras, Jean-Baptiste Holopainen, Ilkka Analysis of PDEs Differential Geometry 35Q55, 35P25, 35J30, 43A85, 43A90 We study the fourth order Schrödinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument. |
| title | Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds |
| topic | Analysis of PDEs Differential Geometry 35Q55, 35P25, 35J30, 43A85, 43A90 |
| url | https://arxiv.org/abs/2105.13804 |