WELL-POSEDNESS OF THE DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS AND THE KLEIN-GORDON EQUATIONS

doi:10.1007/s10473-025-0606-8

Abstract

Abstract: The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrödinger and Klein-Gordon equations. 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 presented in this paper can be summarized as follows:
(1) Discrete Nonlinear Schrödinger Equation: Global well-posedness in $l^p$ spaces for all $1\leq p\leq \infty$, regardless of whether it is in the defocusing or focusing cases.
(2) Discrete Klein-Gordon Equation: Local well-posedness in $l^p$ spaces for all $1\leq p\leq \infty$. Furthermore, in the defocusing case, we establish global well-posedness in $l^p$ spaces for any $2\leq p\leq 2\sigma+2$ ($\sigma>0$). In contrast, in the focusing case, we show that solutions with negative energy blow up within a finite time.
These conclusions reveal the distinct dynamic behaviors exhibited by the solutions of the equations in discrete settings compared to their continuous setting. Additionally, they illuminate the significant role that discretization plays in preventing ill-posedness, and collapse for the nonlinear Schrödinger equation.

Key words: discrete nonlinear Klein-Gordon equation, discrete nonlinear Schrö, dinger equation, well-posedness, blow up, $l^p$

CLC Number:

81Q05

Yifei WU, Zhibo YANG, Qi ZHOU. WELL-POSEDNESS OF THE DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS AND THE KLEIN-GORDON EQUATIONS[J].Acta mathematica scientia,Series B, 2025, 45(6): 2447-2477.

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