Boundary value problems for some quasilinear parabolic equations under minimal conditions on the coefficients

Abstract

Quasilinear parabolic boundary value problems are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness and consistency conditions on the data and some inequalities ensuring the maximum principle for a solution. The proof relies on a priori bounds and the method of continuation in a parameter.