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 Correspondence should be addressed to Changchun Liu, [email protected] Received 31 October 2011; Accepted 6 December 2011 Academic Editor: Hui-Shen Shen Copyright q 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. The existence of weak solutions is studied to the initial Dirichlet problem of the equation ut udiv |∇u| 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.