Existence of renormalized solution for a class of doubly nonlinear parabolic equations with nonstandard growth

We prove the existence of a renormalized solution to a class of doubly nonlinear parabolic equation partial derivative b(xu)/partial derivative t - div (a(x, t, u, del u) + Phi(u)) = f - div(F) in Q, where -diva(x, t, u, del u) is a Leray-Lions operator which is coercive and which grows like vertical bar del u vertical bar(p(x)-1) with respect to del u, but which is not restricted by any growth condition with respect to u and where b(x, u) is an C-1(R)-function strictly increasing with respect u. The data f, F and u(0) respectively belong to L-1(Q), (L-p'((center dot))(Q))(N) and L-1(Omega).