Abstract: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 $\phi(-\Delta)$-type operator defined via a Bernstein function $\phi$. 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 $\phi$-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 $\phi(-\Delta)$-framework for handling fractional dispersive equations with nonlocality.
| Comments: | 34 pages |
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 26A33, 35R11 |
| Cite as: | arXiv:2507.04411 [math.AP] |
| (or arXiv:2507.04411v2 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2507.04411 arXiv-issued DOI via DataCite |
From: Yong Zhen Yang [view email]
[v1]
Sun, 6 Jul 2025 14:55:25 UTC (22 KB)
[v2]
Fri, 22 May 2026 08:41:58 UTC (23 KB)
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