On higher differentiability of solutions of parabolic systems with discontinuous coefficiente and $(p,q)$-growth

We consider weak solutions to parabolic systems of the type where the function a(x, t, ξ) satisfies (p, q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives are contained in the class , where the integrability exponents are coupled by for some κ (0,1). For the gap between the two growth exponents we assume [CDATA[2 ≤ p Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy-Dirichlet problems with the mentioned higher differentiability property.