Entropy solutions for nonlinear parabolic problems involving the generalized $p(x)$-Laplace operator and $L^{1}$ data

Abstract

In this paper we prove the existence of an entropy solution to nonlinear parabolic equations with nonhomogeneous Neumann boundary conditions and initial data in $L^{1}.$ By a time discretization technique we analyze the existence, the uniqueness and the stability questions. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.