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

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.