A complete characterization of extinction versus positivity of solutions to a parabolic problem of p-Laplacian type in graphs

This work is concerned with the extinction and positivity properties of solutions to the discrete p-Laplacian parabolic equation ut=Δp,ωu−f(u), p>1 with a nonlinear absorption on a weighted graph. We provide a complete characterization for f to see when we have an extinctive solution or a positive solution. In addition, the extinction time is estimated for the extinctive solutions, and the decay rate is given to the positive solutions. Finally, we give several numerical examples which illustrate the main results.