Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

The chemotaxis-growth system is considered under homogeneous Neumann boundary conditions in smoothly bounded domains , . For any choice of , the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the re...

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Bibliographic Details
Published inAdvanced nonlinear studies Vol. 20; no. 4; pp. 795 - 817
Main Author Winkler, Michael
Format Journal Article
LanguageEnglish
Published De Gruyter 01.11.2020
Subjects
35B40
35K55
35Q92
92C17
Chemotaxis
Logistic Source
Stabilization
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ISSN1536-1365
2169-0375
DOI10.1515/ans-2020-2107

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Summary:The chemotaxis-growth system is considered under homogeneous Neumann boundary conditions in smoothly bounded domains , . For any choice of , the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ( ), the present work shows that, whenever , under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state in the large time limit.
ISSN:1536-1365
2169-0375
DOI:10.1515/ans-2020-2107