Extinction and positivity of solutions of the p-Laplacian evolution equation on networks

In this paper, we consider a discrete version of the following p-Laplacian evolution equation u t = Δ p u , with p > 1 , on a network. Using the discrete p-Laplacian operator in graphs which means a nonlinear diffusion phenomenon on networks, we first introduce the p-Laplacian evolution equation on networks. In fact, spatial derivative in p-Laplacian evolution equation, comparing the continuous case, is replaced with the discrete p-Laplacian operator. Thus, the resulting system is semi-discrete, discrete in space and continuous in time. The main concern is to investigate the large time behaviors of nontrivial solutions of this equation whose initial data are nonnegative and the boundary data vanish. In order to do so, we use the analytic approaches such as vector calculus on networks, maximum principle and comparison principle instead of numerical ones. After deriving the basic properties of this equation, we finally prove that the solution becomes extinct for 1 < p < 2 and remains strictly positive for p ⩾ 2 .