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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.14883 |
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| _version_ | 1866911275284430848 |
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| author | Aigner, Bernhard Waurick, Marcus |
| author_facet | Aigner, Bernhard Waurick, Marcus |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Evolutionary equations with state-dependent delay Aigner, Bernhard Waurick, Marcus Analysis of PDEs 35A01, 35A02 (Primary) 34K43, 35G40, 47A50, 47B02 (Secondary) 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. |
| title | Evolutionary equations with state-dependent delay |
| topic | Analysis of PDEs 35A01, 35A02 (Primary) 34K43, 35G40, 47A50, 47B02 (Secondary) |
| url | https://arxiv.org/abs/2511.14883 |