Journal of Applied Mathematics
Volume 2012 (2012), Article ID 835495, 21 pages
http://dx.doi.org/10.1155/2012/835495
Research Article

Existence of Solutions for the Evolution 𝑝 ( 𝑥 ) -Laplacian Equation Not in Divergence Form

Changchun Liu, Junchao Gao, and Songzhe Lian

Department of Mathematics, Jilin University, Changchun 130012, China

Received 31 October 2011; Accepted 6 December 2011

Academic Editor: Hui-Shen Shen

Copyright © 2012 Changchun Liu 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

The existence of weak solutions is studied to the initial Dirichlet problem of the equation 𝑢 𝑡 = 𝑢 d i v ( | ∇ 𝑢 | 𝑝 ( 𝑥 ) − 2 ∇ 𝑢 ) , with inf 𝑝 ( 𝑥 ) > 2 . We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.