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# Non-autonomous evolutionary equations

Citation Link: https://doi.org/10.15480/882.4182

Publikationstyp

Book part

Publikationsdatum

2022

Sprache

English

Institut

First published in

Number in series

287

Start Page

259

End Page

273

Citation

Operator Theory: Advances and Applications 287: 259-273 (2022)

Publisher DOI

Scopus ID

Publisher

Springer

Previously, we focussed on evolutionary equations of the form (∂t,νM(∂t,ν)+A¯)U=F. $$\displaystyle \left (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\right )U=F. $$ In this chapter, where we turn back to well-posedness issues, we replace the material law operator M(∂t,ν), which is invariant under translations in time, by an operator of the form ℳ+∂t,ν−1N, $$\displaystyle \mathcal {M}+\partial _{t,\nu }^{-1}\mathcal {N}, $$ where both ℳ and N are bounded linear operators in L2,ν(ℝ; H). Thus, it is the aim in the following to provide criteria on ℳ and N under which the operator ∂t,νℳ+N+A $$\displaystyle \partial _{t,\nu }\mathcal {M}+\mathcal {N}+A $$ is closable with continuous invertible closure in L2,ν(ℝ; H). In passing, we shall also replace the skew-selfadjointness of A by a suitable real part condition. Under additional conditions on ℳ and N, we will also see that the solution operator is causal. Finally, we will put the autonomous version of Picard’s theorem into perspective of the non-autonomous variant developed here.

DDC Class

510: Mathematik

Publication version

publishedVersion

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