Local/global well-posedness analysis of time-space fractional Schrödinger equation on $\mathbb{R}^{d}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917521200775168 |
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| author | Yang, Yong Zhen Zhou, Yong |
| author_facet | Yang, Yong Zhen Zhou, Yong |
| contents | We investigate a class of nonlinear time-space fractional Schrödinger equations with nonlocal effects in both time and space. The time derivative is of Achar type, and the space operator is a $ϕ(-Δ)$-type operator defined via a Bernstein function $ϕ$. This nonlocality invalidates classical Strichartz estimates. By combining asymptotic analysis of Mittag-Leffler functions, the Hörmander multiplier theorem, and harmonic analysis techniques, we establish a Gagliardo-Nirenberg inequality in $ϕ$-Triebel-Lizorkin spaces and derive key Sobolev estimates for the solution operator. These analyses yield the local and global well-posedness of the equations in appropriate Banach spaces. Our work demonstrates the effectiveness of the $ϕ(-Δ)$-framework for handling fractional dispersive equations with nonlocality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local/global well-posedness analysis of time-space fractional Schrödinger equation on $\mathbb{R}^{d}$ Yang, Yong Zhen Zhou, Yong Analysis of PDEs 26A33, 35R11 We investigate a class of nonlinear time-space fractional Schrödinger equations with nonlocal effects in both time and space. The time derivative is of Achar type, and the space operator is a $ϕ(-Δ)$-type operator defined via a Bernstein function $ϕ$. This nonlocality invalidates classical Strichartz estimates. By combining asymptotic analysis of Mittag-Leffler functions, the Hörmander multiplier theorem, and harmonic analysis techniques, we establish a Gagliardo-Nirenberg inequality in $ϕ$-Triebel-Lizorkin spaces and derive key Sobolev estimates for the solution operator. These analyses yield the local and global well-posedness of the equations in appropriate Banach spaces. Our work demonstrates the effectiveness of the $ϕ(-Δ)$-framework for handling fractional dispersive equations with nonlocality. |
| title | Local/global well-posedness analysis of time-space fractional Schrödinger equation on $\mathbb{R}^{d}$ |
| topic | Analysis of PDEs 26A33, 35R11 |
| url | https://arxiv.org/abs/2507.04411 |