The Existence of Entropy Solutions for a Class of Parabolic Equations

The existence and uniqueness of entropy solutions for a class of parabolic equations involving a p(x)-Laplace operator are investigated. We first prove existence of the global weak solution for the p(x)-Laplacian equations with regular initial data via the difference and variation methods as well as the standard domain expansion technique. Then, by constructing and solving a related approximation problem, the entropy solution for the p(x)-Laplacian equations with irregular initial data in whole space is also obtained.