Evolutionary equations with state-dependent delay
Bernhard Aigner, Marcus Waurick
TL;DR
The work addresses PDEs with state-dependent delay by extending a contraction-mapping approach from ODEs to the Picard evolutionary-equation setting, using exponentially weighted spaces to obtain weak-solution well-posedness. It develops a distributional reformulation on the full line alongside a projection technique to handle half-line initial data, requiring both a regular right-hand side and a consistency condition. The results cover a broad class of material laws (including simple laws), establishing local (and in some cases global) existence, uniqueness, and continuous dependence, with explicit treatment of heats, waves, Maxwell, reaction–diffusion, fractional derivatives, convolutions, and port-Hamiltonian systems. This yields a versatile framework for nonlocal-in-time PDEs with state-dependent delays, extending beyond semigroup theory by leveraging the evolutionary-equation perspective and regularity-increasing right-hand sides. The approach offers a robust toolset for modeling complex delay dynamics in continuum systems with potential applications in physics and engineering.
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.