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Well-posedness of quasilinear parabolic equations in time-weighted spaces

Part of: Equations of mathematical physics and other areas of application Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  26 November 2024

Bogdan-Vasile Matioc Open the ORCID record for Bogdan-Vasile Matioc [Opens in a new window]  and
Christoph Walker Open the ORCID record for Christoph Walker [Opens in a new window]
Bogdan-Vasile Matioc
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany (bogdan.matioc@ur.de)
Christoph Walker
Affiliation:
Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany (walker@ifam.uni-hannover.de) (corresponding author)
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Abstract

Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.

Keywords

quasilinear parabolic problemsemiflowwell-posednesstime-weighted-spaceschemotaxis equations

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35K58: Semilinear parabolic equations 35K59: Quasilinear parabolic equations 35Q92: PDEs in connection with biology and other natural sciences

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 33
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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