Boundary Value Problems for Some Quasilinear Parabolic Equations Under Minimal Conditions on the Coefficients

Quasilinear parabolic boundary value problems are considered. Sharp conditions on the data ensuring the existence and uniqueness of solutions in Sobolev classes Wp1,2(Q)(p>(n+2)/2 are exposed. They are smoothness and consistency conditions on the data and some inequalities ensuring the maximum principle for a solution. The proof of a priori estimates employs the method of frozen coefficients. The proof of existence relies on a priori bounds and Leray–Schauder-type continuation.