Extremal solutions of quasilinear parabolic subdifferential inclusions

Abstract

In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by general operators of Leray-Lions type and continuously perturbed subdifferentials. Our main goal is to show the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. One of the main difficulties that arises in the treatment of the parabolic problems considered here is due to the fact that the underlying solution space does not possess lattice structure. Furthermore, the variational techniques that can be used for the corresponding stationary problems are not applicable here. The main tools used in the proof of our result are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques.