Pattern formations and spatial entropy for spatially discrete diffusion equations

Formation of mosaic patterns for spatially discrete diffusion equations with cubic nonlinearity is investigated. We construct feasible basic patterns in each parameter region and combine these basic patterns into large patterns on one- and two-dimensional lattices. The basic patterns are characterized and constructed through formulating parameter conditions based on a geometrical setting. Spatial entropy associated with these patterns are computed or estimated. We also consider three typical boundary conditions and investigate their influences on pattern formations and spatial entropy. Several numerical computations are performed to illustrate such a formation of patterns.