Global solutions to 3D quadratic nonlinear Schrödinger-type equation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918286910816256 |
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| author | Guo, Zihua Liu, Naijia Song, Liang |
| author_facet | Guo, Zihua Liu, Naijia Song, Liang |
| contents | We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us more flexibilities in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10804 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global solutions to 3D quadratic nonlinear Schrödinger-type equation Guo, Zihua Liu, Naijia Song, Liang Analysis of PDEs 35Q55, 35E15 We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us more flexibilities in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem. |
| title | Global solutions to 3D quadratic nonlinear Schrödinger-type equation |
| topic | Analysis of PDEs 35Q55, 35E15 |
| url | https://arxiv.org/abs/2409.10804 |