Evolutionary equations with state-dependent delay

TL;DR

This paper extends contraction mapping methods to prove local well-posedness for a broad class of evolutionary PDEs with state-dependent delays, including classical and fractional equations, under certain regularity conditions.

Contribution

It generalizes contraction mapping techniques to evolutionary PDEs with state-dependent delays, covering a wide range of classical and modern equations.

Findings

Established local well-posedness for generalized initial value problems.

Applied results to heat, wave, Maxwell, and fractional PDEs.

Demonstrated the approach's applicability to various systems in mathematical physics.

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 $\bigl(\partial_{t} M(\partial_{t}) + A\bigr) u(t) = F\bigl(t,u_{(t)}\bigr)$, 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.