A new contribution to the interior differentiability for nonlinear variational parabolic systems with nonlinearity q greater or equal two

In this work we study the interior differentiability in $ x$ and $ t$ for the weak solutions $$ u\in L^{q}(-T,0,H^{1,q}(\Omega,\mathbb R^{N}))\cap C^{0,\lambda}(\,{\!Q},\mathbb R^{N})$$ $(q\ge 2,\,0<\lambda<1,\,Q=\Omega\times]-T,0[\subset\mathbb R^{n+1},\,N\, {\it integer}\, >1)$ to the second order nonlinear parabolic system in divergence form $$ -\sum_{j=1}^{n}D_{j}a^{j}(X,u,Du)+\frac{\partial u}{\partial t}= B^{0}(X,u,Du)\;,\quad X=(x,t)\in Q\,.$$ Suitable hypotheses of ${q}$-nonlinearity $(q\ge 2)$ and strict monotonicity on the coefficients are adopted.