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

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 $l_h^p$ spaces for all $1\le p\le \mathrm{\infty }$, 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 $l_h^p$ spaces for all $1\le p\le \mathrm{\infty }$. Furthermore, in the defocusing case, we establish global well-posedness in $l_h^p$ spaces for any $2\le p\le 2\sigma +2$. In contrast, in the focusing case, we show that solutions with negative energy blow up within a finite time.