Local/global well-posedness analysis of time-space fractional Schr\"{o}dinger equation on $\mathbb{R}^{d}$

Yong Zhen Yang; Yong Zhou

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arXiv · 2507.04411

Local/global well-posedness analysis of time-space fractional Schr\"{o}dinger equation on $\mathbb{R}^{d}$

Yong Zhen Yang·Yong Zhou

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Abstract

We investigate a class of nonlinear time-space fractional Schr\"{o}dinger 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 H\"{o}rmander 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.

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BibTeX RIS

Yong Zhen Yang, Yong Zhou. 2025-07-06. Local/global well-posedness analysis of time-space fractional Schr\"{o}dinger equation on $\mathbb{R}^{d}$. https://arxiv.org/abs/2507.04411

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