Extremal solutions of quasilinear parabolic inclusions with generalized Clarkeʹs gradient

In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarkeʹs generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray–Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarkeʹs gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis.