Extinction and Positivity of the Solutions for a p -Laplacian Equation with Absorption on Graphs

Abstract

We deal with the extinction of the solutions of the initial-boundary value problem of the discrete p-Laplacian equation with absorption $u_t={\Delta }_{p,\omega }u-u^q$ with $p>1$, $q>0$, which is said to be the discrete p-Laplacian equation on weighted graphs. For $0<q<1$, we show that the nontrivial solution becomes extinction in finite time while it remains strictly positive for $p\ge 2$, $q\ge 1$ and $q\ge p-1$. Finally, a numerical experiment on a simple graph with standard weight is given.