Control and stabilization problem for a class of fourth-order nonlinear Schrödinger equation on boundaryless compact manifold
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911284695400448 |
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| author | Song, Yilin Zheng, Jiqiang Zhou, Ruihan |
| author_facet | Song, Yilin Zheng, Jiqiang Zhou, Ruihan |
| contents | In this paper, we study the control and stabilization problem for a class of fourth-order Schrödinger equation on compact manifold without boundary with dimensions $d\in[1,5]$:
\begin{align*}
i\partial_tu+(Δ_g^2-βΔ_g)u=|u|^{2k}u,
\end{align*}
where $k\in\Bbb N$.
For $1\leq d\leq4$ and $k\geq1$, we combine the method proposed by Loyola and semiclassical analysis to prove the stabilization result only under the geometric control condition (GCC), which removes the unique continuation assumption in Capistrano-Filho-Pampu [Math. Z. (2022)]. For $d=5$, we focus on a special case, i.e. $\Bbb S^5$. Establishing the propagation of singularity in Bourgain space, we prove the similar control and stabilization result in energy space as lower dimensions, which generalizes the result of Laurent [SIAM J. Math. Anal. (2009)]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19890 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Control and stabilization problem for a class of fourth-order nonlinear Schrödinger equation on boundaryless compact manifold Song, Yilin Zheng, Jiqiang Zhou, Ruihan Analysis of PDEs 35Q55 In this paper, we study the control and stabilization problem for a class of fourth-order Schrödinger equation on compact manifold without boundary with dimensions $d\in[1,5]$: \begin{align*} i\partial_tu+(Δ_g^2-βΔ_g)u=|u|^{2k}u, \end{align*} where $k\in\Bbb N$. For $1\leq d\leq4$ and $k\geq1$, we combine the method proposed by Loyola and semiclassical analysis to prove the stabilization result only under the geometric control condition (GCC), which removes the unique continuation assumption in Capistrano-Filho-Pampu [Math. Z. (2022)]. For $d=5$, we focus on a special case, i.e. $\Bbb S^5$. Establishing the propagation of singularity in Bourgain space, we prove the similar control and stabilization result in energy space as lower dimensions, which generalizes the result of Laurent [SIAM J. Math. Anal. (2009)]. |
| title | Control and stabilization problem for a class of fourth-order nonlinear Schrödinger equation on boundaryless compact manifold |
| topic | Analysis of PDEs 35Q55 |
| url | https://arxiv.org/abs/2511.19890 |