A critical exponent in a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects
type: Item
instance of: scholarly article
title: A critical exponent in a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects
zbMATH Open document ID: 1544.35053
author: Christian Stinner
author: Michael Winkler
published in: Journal of Evolution Equations
publication date: 2024-07-10T00:00:00Z
review text: In this paper, the authors study a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects endowed with homogeneous Neumann boundary conditions in a smoothly bounded domain \(\mathbb{R}^{n}\), \(n\geq3\) with sufficiently regular functions \(D(u)\) and \(S(u)\). When \(\frac{S(u)}{D(u)}\cong Cu^{\alpha}\) with \(C>0\), they establish some criticality of the value \(\alpha=\frac{2}{n}\) for \(n\geq3\), without any additional assumption on the behavior of \(D(s)\) as \(s\rightarrow\infty\), in particular without requiring any algebraic lower bound for \(D\). Moreover, applied to the Keller-Segel system with volume-filling effect for probability distribution functions of the type \(Q(s)=\exp(-s^{\beta})\), \(s\geq0\), for global solvability the exponent \(\beta=\frac{n-2}{n}\) is seen to be critical.\N\NThese results are undoubtedly novel and intriguing, and they extend previous understanding in a seamless manner. The derivation is presented and organized exceptionally well, building upon existing ideas from the literature and appropriately citing the relevant references. Moreover, the work introduces significant and nontrivial new ideas. Overall, in my opinion, this is an exemplary piece of research.
reviewed by: Pan Zheng
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 7879408
zbMATH Keywords: chemotaxis
zbMATH Keywords: quasilinear Keller-Segel system
zbMATH Keywords: global weak energy solution
zbMATH Keywords: critical exponent
zbMATH Keywords: volume-filling effect
MaRDI profile type: Publication
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DOI: 10.1007/S00028-024-00954-X
description: scientific article; zbMATH DE number 7879408