  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.
