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

In this paper, we study the existence of solutions for nonlinear parabolic initial boundary value problem with p(x)p(x)-growth conditions in the space W1,xLp(x)(Q)∩L∞(0,T;L2(Ω))W1,xLp(x)(Q)∩L∞(0,T;L2(Ω)), and give an existence theorem of weak solutions for the following equation∂u∂t+A(u)=f, where A(u)=−diva(x,t,u,∇u)+a0(x,t,u,∇u), a(x,t,u,∇u)a(x,t,u,∇u) and a0(x,t,u,∇u)a0(x,t,u,∇u) satisfy p(x)p(x)-growth conditions with respect to u and ∇u  , and f∈W−1,xLq(x)(Q)f∈W−1,xLq(x)(Q).