Evolutionary equations with state-dependent delay

Abstract

We extend a contraction mapping argument for ordinary state-dependent delay differential equations to evolutionary partial differential equations in the sense of R. Picard, that is, to equations of the form $({\partial }_{t}M({\partial }_t)+A)u(t)=F(t,u_{(t)})$, where $A$ is an $\mathrm{m}$-accretive (unbounded) linear operator and $M$ is a material law. We establish local well-posedness (in the sense of weak solutions) of generalized initial value problems that stem from a distributional formulation. We require prehistories in $H^1$ with bounded derivative, a regularity increasing right-hand side and a consistency condition. We showcase the viability of our results by applying them to classical examples (heat, wave and Maxwell's equations), examples from semigroup theory, port-Hamiltonian systems, as well as equations featuring fractional derivatives and convolutions (in time) with bounded operators.