Existence of solutions for nonlinear parabolic problem with -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 ; L 2 ( Ω ) ) , and give an existence theorem of weak solutions for the following equation ∂ u ∂ t + A ( u ) = f , where A ( u ) = − div a ( x , t , u , ∇ u ) + a 0 ( x , t , u , ∇ u ) , a ( x , t , u , ∇ u ) and a 0 ( x , t , u , ∇ u ) satisfy p ( x ) -growth conditions with respect to u and ∇u, and f ∈ W − 1 , x L q ( x ) ( Q ) .