Extinction and positivity of the solutions of the heat equations with absorption on networks

In this paper, we propose a discrete version of the following semilinear heat equation with absorption u t = Δ u − u q with q > 1 , which is said to be the ω-heat equation with absorption on a network. Using the discrete Laplacian operator Δ ω on a weighted graph, we define the ω-heat equations with absorption on networks and give their physical interpretations. The main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish. It is proved that the asymptotic behaviors of the solutions u ( x , t ) as t tends to +∞ strongly depend on the sign of q − 1 .