Interior Differentiability Results for Nonlinear Variational Parabolic Systems with Nonlinearity q in ]1,2[

In this work we prove that the solutions ${\scriptstyle u\in L^{q}(-T,0,H^{1,q}(\Omega,\RR^{N}))\cap C^{0,\lambda}(\overline{\!Q},\RR^{N})}$ ${\scriptstyle (1<q<2,\,0<\lambda<1,\,Q=\Omega\times]-T,0[,\,N\, \hbox{\piccoloit integer}\, >1)}$ to the second order nonlinear parabolic system of variational type $${\scriptstyle -\sum_{j=1}^{n}D_{j}a^{j}(X,u,Du)+{\partial u\over\partial t}= B^{0}(X,u,Du)\;,\quad X=(x,t)\in Q\,,}$$ belong to the space ${\scriptstyle L^{q}(-a,0,H^{2,q}_{\loc}(\Omega,\RR^{N}))\cap H^{1,q}(-a,0,L^{q}_{\loc}(\Omega,\RR^{N}))}$, for all ${\scriptstyle a\in]0,T[}$, provided suitable hypotheses of ${\scriptstyle q}$-nonlinearity ${\scriptstyle (1<q<2)}$ on the coefficients are adopted.