Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908331652677632 |
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| author | Toprak, Ebru Xie, Mengyi |
| author_facet | Toprak, Ebru Xie, Mengyi |
| contents | We prove small data scattering for the fourth-order Schrödinger equation with quadratic nonlinearity \begin{equation*}
i\partial_t u+Δ^2 u+αu^2 + β\bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $α, β\in \mathbb{R}$. We extend the space-time resonance method, originally introduced by Germain, Masmoudi, and Shatah, to the setting involving the bilaplacian. We show that under a smallness condition on the initial data measured in a suitable norm, the solution satisfies $\|u\|_{L^{\infty}_x }\lesssim t^{-\frac{5}{4}} $ and scatters to the solution to the free equation. Although our work builds upon an established method, the fourth-order nature of the equation presents substantial challenges, requiring different techniques to overcome them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations Toprak, Ebru Xie, Mengyi Analysis of PDEs We prove small data scattering for the fourth-order Schrödinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+Δ^2 u+αu^2 + β\bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $α, β\in \mathbb{R}$. We extend the space-time resonance method, originally introduced by Germain, Masmoudi, and Shatah, to the setting involving the bilaplacian. We show that under a smallness condition on the initial data measured in a suitable norm, the solution satisfies $\|u\|_{L^{\infty}_x }\lesssim t^{-\frac{5}{4}} $ and scatters to the solution to the free equation. Although our work builds upon an established method, the fourth-order nature of the equation presents substantial challenges, requiring different techniques to overcome them. |
| title | Global Solutions for 5D Quadratic Fourth-Order Schrödinger Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.15572 |