${\displaystyle L_{\infty }}$-estimate for solutions of nonlinear parabolic systems

We prove existence of weak solutions to nonlinear parabolic systems with p-Laplacians terms in the principal part. Next, in the case of diagonal systems an ${\displaystyle L_{\infty }}$-estimate for weak solutions is shown under additional restrictive growth conditions. Finally, ${\displaystyle L_{\infty }}$-estimates for weakly nondiagonal systems (where nondiagonal elements are absorbed by diagonal ones) are proved. The ${\displaystyle L_{\infty }}$-estimates are obtained by the Di Benedetto methods.