Abstract
This Chapter is devoted to an initial boundary value problem for a parabolic inclusion with a multivalued nonlinearity given by a generalized gradient in the sense of Clarke [13] of some locally Lipschitz function. The elliptic operator is a general quasilinear operator of Leray-Lions type. Of special interest is the case where the multivalued term is described by the usual subdifferential of a convex function. Our main result is the existence of extremal solutions limited by prescribed lower and upper solutions. The main tools used in the proofs are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as the calculus with generalized gradients.KeywordsGeneralize GradientInitial BoundaryMaximal MonotoneNonlinear Evolution EquationExtremal SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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