Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and non linearity $qge2$

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Let $\Omega $ be a bounded open subset of $ R^{n} $, let $X = (x, t)$ be a point of $ R^{n}\times R^{N} $
In the cylinder
$ Q = \Omega \times(-T, 0)$,$ T > 0$, we deduce the local differentiability result
$u \in L^{2}(-a,0,H^{2}(B(\sigma),\mathbb{R}^{N}))\cap H^{1}(-a,o,L^{2}(B(\sigma),\mathbb{R}^{N}))$
for the solutions u of the class
$ L^q(-T, 0,H^{1,q}(\Omega,R^{N})) \cap C^{0,\lambda}(\overline Q,R^{N}))$
$ (0 < \lambda < 1, N $ integer $\geq 1)$
of the nonlinear parabolic system
$$ -\sum_{j=1}^{n}D_{j}a^{j}(X,u,Du)+\frac{\partial u}{\partial t}=
B^{0}(X,u,Du)$$
with quadratic growth and nonlinearity $q \geq 2$. This result had been obtained making
use of the interpolation theory and an embedding theorem of Gagliardo-Nirenberg type
for functions u belonging to $W^{1,q}\cap C^{0,\lambda}$.

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