L\({}^ p\)-regularity and Hölder continuity for solutions of second order parabolic systems (Q914977)

This paper deals with nonlinear parabolic systems of the form \[ u^ i_ t-D_{\alpha}A_ i^{\alpha}(z,u,Du)=B_ i(z,u,Du),\quad i=1,2,...,N, \] or with a special case of this system which corresponds to \(A_ i^{\alpha}\) linear with respect to the derivatives. The systems are considered in the set [0,T]\(\times Q\), with \(Q\subset R^ n\) an open set. Under adequate growth conditions on the functions involved in the system, estimates for the weak solutions are obtained in certain integral norms. Hölder continuity of weak solutions is also obtained in the paper, under suitable conditions. The proofs are based on sharp estimates obtained in several lemmas.