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,.