Well-posedness of the discrete nonlinear Schrödinger equations and the Klein-Gordon equations

2023/10/31 by Wu, Yifei, Yang, Zhibo, Zhou, Qi
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.20382

Abstract

The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrödinger 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 Schrödinger Equation: We establish global well-posedness in l^p_h spaces for all 1≤ p≤ ∞, 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≤ p≤ ∞. Furthermore, in the defocusing case, we establish global well-posedness in l^p_h spaces for any 2≤ p≤ 2σ+2. In contrast, in the focusing case, we show that solutions with negative energy blow up within a finite time.

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