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

Abstract

The existence of weak solutions is studied to the initial Dirichlet problem of the equation 𝑢𝑡=𝑢div(|∇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.