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

We consider weak solutions u : OT. RN to parabolic systems of the type u(t) - div a(x, t, Du) = 0 in OT = Ox (0, T), where the function a(x, t,.) 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 x . a(x, t,.) 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 Dxa(center dot, center dot,.) are contained in the class La( 0, T; L beta( O)), where the integrability exponents a, beta are coupled by p(n + 2) - 2n 2a + n/beta = 1-kappa for some.. (0, 1). For the gap between the two growth exponents we assume. 2 p < q p + 2. n + 2. 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 ut. Lp/(q- 1) loc (OT). We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy-Dirichlet problems with the mentioned higher differentiability property.