Existence of entropy solutions for some strongly nonlinear p(x)-parabolic problems with L-1-data

This paper is devoted to study the following strongly nonlinear p(x)-parabolic problem {partial derivative u/partial derivative t - div a(x, t, del u) + g(x, t, u del u) + delta vertical bar u vertical bar(p(x)-2)u = f - div phi(u) in Q(T), u = 0 on Sigma(T), (1) u(x, 0) = u(0) in Omega, with f is an element of L-1(Q(T)), phi(center dot) is an element of C-0(R, R-N), u(0) is an element of L-1(Omega) and delta > 0. We prove the existence of entropy solutions for this problem in the parabolic space with variable exponent V.