The authors consider the regularity for weak solutions of the following nonlinear parabolic systems with Dirichlet data: \[ (\partial /\partial tu^{(l)}-div(| Du|^{p-2}\cdot Du^{(l)})=f^{(l)}(x,t,u,Du)\quad in\quad \Omega_ T=\Omega \times (0,T], \]
\[ u^{(l)}(\cdot,t)=g^{(l)}(\cdot,t)\quad on\quad \partial \Omega \times \{t\},\quad t\in (0,T]. \] (\(\Omega\) \(\subset {\mathbb{R}}^ N\), \(0<T<\infty\), \(p>\max \{l,2N/(N+2)\}\), \(l=1,2,...,m)\). The basic assumption on f(x,t,u,Du) is: \(| f| \leq C(1+| Du|^{p- 1})\). The Hölder continuity and a priori estimates of the Hölder norm up to the lateral surface of \(\Omega_ T\) are obtained for the weak solution, furthermore for the gradient of the solution if the data are homogeneous ones. The validity of “everywhere” regularity is due to the particular structure of the systems above.
Analogous results for the corresponding elliptic systems are indicated, too.