Semilinear parabolic differential inclusions with one-sided Lipschitz nonlinearities

Abstract

We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets. Preliminaries from set-valued analysisWe refer to the monographs [1] and [18] for general notions from setvalued analysis. In the following we specify some notation that will be used throughout this paper. In the following, let X and Y be normed spaces.Definition 1. For any x ∈ X and any subset M ⊆ X, we define the distance of x to M by dist(x, M ) X = inf{ x − y X : y ∈ M } and the proximal set byRecall that proj(x, M ) X is a singleton in case M is convex and X is a Hilbert space. Moreover, by a common abuse of notation, we write M X := sup x∈M x X for M ⊆ X bounded.By CBC(X) we denote the set of all closed, bounded, and convex subsets of X. There are various ways of defining a topology on CBC(X) which in general are not equivalent. We will use convergence in the Hausdorff and in the Kuratowski sense. Definition 2. For any two sets M, M ⊆ X, the Hausdorff semidistance and the Hausdorff distance are defined by dist(M, M ) X := sup x∈M dist(x, M ) X , dist H (M, M ) X := max{dist(M, M ) X , dist( M , M ) X }.It is well-known that dist H defines a metric on CBC(X).

One Sided Lipschitz Evolution Inclusions in Banach Spaces

“…Notice also [25], where the theory of monotone operators in the evolution (Gelfand) triple is studied, again using compact embedding of reflexive and separable Banach space in a separable Hilbert one. Such a problem, but with many qualitative results, is studied in [22]. In [6], either compactness conditions or Lipschitz continuity in a Banach space with uniformly convex dual are used.…”

Existence of strong solutions for the Oldroyd model with multivalued right-hand side

“…Extensions of the results shown in Deimling [14] can be found in O'Regan [28]. Existence of global solutions to multivalued differential equation with a linear parabolic principal part and a relaxed one-sided Lipschitz nonlinear set-valued operator is, e.g., considered in Beyn, Emmrich, and Rieger [7].…”

On a Multivalued Differential Equation with Nonlocality in Time

“…In O'Regan [32], some extensions of the results shown in Deimling [9] are presented. A semilinear multivalued differential equation with a linear, bounded, and strongly positive operator and a set-valued nonlinear operator is, e.g., considered in Beyn, Emmrich, and Rieger [4].…”