Existence of solutions for nonlinear parabolic problem with p (x)-growth

In this paper, we study the existence of solutions for nonlinear parabolic initial boundary value problem with p (x)-growth conditions in the space W^{1, x} L^{p (x)} (Q) ∩ L^∞ (0, T ; L² (Ω)), and give an existence theorem of weak solutions for the following equationfrac(∂ u, ∂ t) + A (u) = f, where A (u) = - div a (x, t, u, ∇ u) + a₀ (x, t, u, ∇ u), a (x, t, u, ∇ u) and a₀ (x, t, u, ∇ u) satisfy p (x)-growth conditions with respect to u and ∇u, and f ∈ W^{- 1, x} L^{q (x)} (Q).