Journal of Applied Mathematics
Volume 2011 (2011), Article ID 937079, 12 pages
http://dx.doi.org/10.1155/2011/937079
Research Article

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

Qiao Xin, Chunlai Mu, and Dengming Liu

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 1 April 2011; Accepted 4 July 2011

Academic Editor: Alain Miranville

Copyright © 2011 Qiao Xin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 = Δ 𝑝 , 𝜔 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 ≥ 2 , 𝑞 ≥ 1 and 𝑞 ≥ 𝑝 − 1 . Finally, a numerical experiment on a simple graph with standard weight is given.