Parabolic equations with p, q-growth

We consider parabolic equations of the type partial derivative(t)u - div a(x, t, Du) = 0 on a parabolic space time cylinder Omega(T). The vector field a is assumed to satisfy a non-standard p, q-growth assumption. When 2 <= p <= q < p + 4/n it is established that any weak solution u is an element of L-P (0, T; W-1,W-P (Omega)) boolean AND L-loc(q) (0, T;W-loc(1,q)(Omega)) admits a locally bounded spatial gradient Du. Moreover, it is shown that the stronger assumption 2 <= p <= q < p + 4/n+2 guarantees an existence result for the Cauchy-Dirichlet problem associated to the parabolic equation from above. The results cover for example equations of the type partial derivative(t)u - Sigma n=1 partial derivative/partial derivative(xi) ((mu(2) + vertical bar D(i)u vertical bar(2))(pi-2/2) D(i)u) = 0 with mu is an element of [0, 1] and suitable growth exponents p(i). We emphasize that the results include the degenerate case mu = 0. (C) 2013 Elsevier Masson SAS. All rights reserved.