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 $u_t=u\text{div}({\left|\nabla u\right|}^{p(x)-2}\nabla u)$, 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.