Existence of entropy solutions for some strongly nonlinear p(x)-parabolic problems with L1-data

Abstract

This paper is devoted to study the following strongly nonlinear p(x)-parabolic problem

du/d t-  div  a(x,t,\nabla u) + g(x,t,u,\nabla u) + \delta|u|^{p(x)-2}u = f  -  div \phi(u)   in  Q_T,
 u  = 0                      on     \Sigma_{T},
 u(x,0)= u_0             in           \Omega,

with  f \in L^{1}(Q_T),  \phi\in C^{0}(IR,IR^{N}),  u_0 \in L^{1}(\Omega) and  \delta>0. We prove the existence of entropy solutions for this problem in the parabolic space with variable exponent V.