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.