Global existence and blow-up of point-wise solutions to a logarithmic nonlinear heat equation on locally finite graphs

This paper is devoted to studying the logarithmic nonlinear heat equations on a locally finite connected weighted graph G = (V, E). In a discrete sense, the global existence of point-wise solutions and blow-up phenomena at +∞ are obtained under the general conditions. With the addition of the curvature dimension condition CDE′(n, K)(K 0) and other conditions, it is possible to estimate the range of mountain-pass level by using logarithmic Sobolev inequality, then decay estimates of the global point-wise solutions can be given. Finally, several numerical experiments are provided to validate the observed conclusions.