Semilinear Parabolic Differential Inclusions with One-sided Lipschitz Nonlinearities

TL;DR

This paper establishes an existence result for semilinear parabolic differential inclusions with one-sided Lipschitz nonlinearities, using Galerkin discretizations and finite element methods to analyze convergence.

Contribution

It introduces a novel existence proof for such inclusions within Gelfand triples and studies the convergence of Galerkin-based discretizations.

Findings

Existence of solutions proven for the differential inclusion.

Galerkin discretizations converge to the continuous solution set.

Finite element methods are compatible with the problem's structure.

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.