Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and with nonlinearity greater than two

Let
be a bounded open subset of Rn, let X = (x, t) be a point of Rn ×RN.
In the cylinder Q =
× (−T, 0), T > 0, we deduce the local differentiability result
u ∈ L2(−a, 0,H2(B(), RN)) ∩ H1(−a, 0, L2(B(), RN))
for the solutions u of the class Lq(−T, 0,H1,q(
, RN)) ∩ C0,(¯Q , RN) (0 < < 1, N
integer ≥ 1) of the nonlinear parabolic system
−
n X
i=1
Diai(X, u,Du) +
@u
@t
= B0(X, u,Du)
with quadratic growth and nonlinearity q ≥ 2. This result had been obtained making
use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type
for functions u belonging to W1,q ∩ C0,.