On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916164625498112 |
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| author | Lee, Zachary Yu, Xueying |
| author_facet | Lee, Zachary Yu, Xueying |
| contents | In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schrödinger equations in any dimension $d$ of the following forms, $$i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u = V(t, x) u, \quad \text{and} \quad i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u + F (u, \bar{u}) = 0.$$ We show that a linear solution $u$ with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, if the difference between two nonlinear solutions $u_1$ and $u_2$ decays sufficiently fast at two different times, it implies that $u_1 \equiv u_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07355 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations Lee, Zachary Yu, Xueying Analysis of PDEs In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schrödinger equations in any dimension $d$ of the following forms, $$i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u = V(t, x) u, \quad \text{and} \quad i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u + F (u, \bar{u}) = 0.$$ We show that a linear solution $u$ with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, if the difference between two nonlinear solutions $u_1$ and $u_2$ decays sufficiently fast at two different times, it implies that $u_1 \equiv u_2$. |
| title | On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2208.07355 |