---
author:
- 'Sascha Trostorff[^1]'
title: 'Well-posedness for a general class of differential inclusions'
---

**Abstract.** We consider an abstract class of differential inclusions, which covers differential-algebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those differential inclusions in a pure Hilbert space setting. Moreover, we study the causality of the associated solution operator. The theory is illustrated by an application to a semistatic quasilinear variant of Maxwell’s equations.

**Keywords:** differential inclusions, maximal monotone relations, non-autonomous problems, causality, Maxwell’s equations ****

**2010 MSC:** 34G25, 46N20, 35F60 ****

Introduction 
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In [@Picard] it was shown that most (if not all) linear autonomous equations in mathematical physics share a common form, namely $$(\partial_{t}\mathcal{M}+\mathcal{A})U=F,$$ where $\partial_{t}$ stands for the derivative with respect to time, $\mathcal{M}$ is a suitable bounded operator in space-time, which commutes with $\partial_{t}$ and $\mathcal{A}$ is a skew-selfadjoint operator in space, which in applications is a spatial differential operator. This result was generalised by the author in [@Trostorff2012_NA; @Trostorff2012_nonlin_bd; @Trostorff_2011] to the case of differential inclusions, where the operator $\mathcal{A}$ is replaced by a maximal monotone relation. The resulting problem then takes the form $$(U,F)\in\partial_{t}\mathcal{M}+\mathcal{A}.$$ This generalisation 