Compact embedding for p(x, t)-Sobolev spaces and existence theory to parabolic equations with p(x, t)-growth

In this paper, we establish the compact embedding of p(x, t)-Sobolev spaces into p(x, t)-Lebesgue spaces. Moreover, we prove some existence results for nonlinear parabolic problems of the form partial derivative(t)u - div a(x, t, Du) = f - div (vertical bar F vertical bar F-p(x,F- t)-2) in Omega(T), where the vector-field a(x, t, .) satisfies certain p(x, t)-growth conditions.