Monotone methods for semilinear parabolic and elliptic equations on graphs

This paper is devoted to investigate the extinction and propagation properties of solutions to the graph Laplacian parabolic problems with Kpp type or Allen-Cahn type forcing terms on graphs. To this end, we establish the (strong) maximum principle and the upper and lower solutions method for parabolic and elliptic problems on graphs. The stability of equilibrium solutions is studied by constructing suitable upper and lower solutions. Moreover, we give an example and numerical experiments to demonstrate one of our main results.