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arXiv · 2310.20382

Well-posedness of the discrete nonlinear Schr\"odinger equations and the Klein-Gordon equations

Yifei Wu·Zhibo Yang·Qi Zhou
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Abstract

The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schr\"odinger equation and Klein-Gordon equation. These theories encompass both local and global well-posedness, as well as the existence of blowing-up solutions for large and irregular initial data. The main results of this paper presented in this paper can be summarized as follows: 1. Discrete Nonlinear Schr\"odinger Equation: We establish global well-posedness in $l^p_h$ spaces for all $1\leq p\leq \infty$, regardless of whether it is in the defocusing or focusing cases. 2. Discrete Klein-Gordon Equation (including Wave Equation): We demonstrate local well-posedness in $l^p_h$ spaces for all $1\leq p\leq \infty$. Furthermore, in the defocusing case, we establish global well-posedness in $l^p_h$ spaces for any $2\leq p\leq 2\sigma+2$. In contrast, in the focusing case, we show that solutions with negative energy blow up within a finite time.

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Yifei Wu, Zhibo Yang, Qi Zhou. 2023-10-31. Well-posedness of the discrete nonlinear Schr\"odinger equations and the Klein-Gordon equations. https://arxiv.org/abs/2310.20382

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