*(English)*Zbl 0796.35089

The authors consider quasilinear parabolic equations with principal part in divergence form of the type

in ${\mathcal{D}}^{\text{'}}\left({{\Omega}}_{T}\right)$ where ${\Omega}$ is a bounded open set in ${\mathbb{R}}^{N}$, $0<T<\infty $, ${{\Omega}}_{T}={\Omega}\times (0,T)$; here the functions $a$ and $b$ are assumed to be measurable and to satisfy several further (structure) conditions. Utilizing and generalizing results of O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Ural’tzeva as well as of E. Di Benedetto, the authors establish interior and boundary Hölder estimates for bounded weak solutions, e.g., for suitable Dirichlet and Neumann problems. [For related investigations, cf. also papers by *A. V. Ivanov* of the last five years, e.g., Algebra Anal. 3, No. 2, 139-179 (1991; Zbl 0764.35026)].

##### MSC:

35K65 | Parabolic equations of degenerate type |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35K60 | Nonlinear initial value problems for linear parabolic equations |

35K55 | Nonlinear parabolic equations |

35B45 | A priori estimates for solutions of PDE |