We investigate quasilinear discrete PDEs ∂ t u = Δ N φ (u) + K f (u) ₜ u = N (u) + Kf (u) of reaction-diffusion type with nonlinear diffusion term defined on an n n -dimensional unit torus discretized with mesh size 1 N 1N for N ∈ N N {N}, where Δ N N is the normalized discrete Laplacian, φ is a strictly increasing C 5 C⁵ function and f f is a C 1 C¹ function. We establish L ∞ L^ bounds and space-time Hölder estimates, both uniform in N N, of the first and second spatial discrete derivatives of the solutions. In the equation, K > 0 K>0 is a large constant and we show how these estimates depend on K K
Funaki et al. (Wed,) studied this question.