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

\documentclass{article} \usepackage{amsmath,amssymb} \begin{document} 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 $W^{1,q}\cap C^{0,\lambda}$. \end{document}