  In this paper we develop a geometric theory for quasilinear parabolic
problems in weighted $L_p$-spaces. We prove existence and uniqueness of
solutions as well as the continuous dependence on the initial data. Moreover,
we make use of a regularization effect for quasilinear parabolic equations to
study the $\om$-limit sets and the long-time behaviour of the solutions. These
techniques are applied to a free boundary value problem. The results in this
paper are mainly based on maximal regularity tools in (weighted) $L_p$-spaces.
