Existence of weak solutions of parabolic systems with p, q-growth

We consider evolutionary problems associated with a convex integrand , which is -Holder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions , which solve partial derivative(t)u - div partial derivative(zeta) integral(x, t, Du) = 0 weakly in Omega(T). Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q - p, namely 2n/n + 2 < p <= q < p + 1. For 2n/n + 2 < p <= q < p + min{2, p}alpha/n + 2, we prove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.