The well-posedness of generalized nonlinear wave equations on the lattice graph

TL;DR

This paper develops a new derivative concept on lattice graphs, extending nonlinear wave equation theory to these structures, and proves well-posedness results for various types of wave equations.

Contribution

Introduces a novel first-order derivative on lattice graphs and establishes well-posedness results for generalized nonlinear wave equations in this setting.

Findings

Established weak (1,1) and strong (p,p) estimates for the new derivative

Proved local and long-time well-posedness for quasilinear wave equations

Proved global well-posedness for defocusing semilinear wave equations

Abstract

In this paper, we introduce a novel first-order derivative for functions on a lattice graph, and establish its weak (1, 1) estimate as well as strong (p, p) estimate for p > 1 in weighted spaces. This derivative is designed to reconstruct the discrete Laplacian, enabling an extension of the theory of nonlinear wave equations, including quasilinear wave equations, to lattice graphs. We prove the local well-posedness of generalized quasilinear wave equations and the long-time well-posedness of these equations for small initial data. Furthermore, we prove the global well-posedness of defocusing semilinear wave equations for large initial data.