Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
type: Item
instance of: scholarly article
title: Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
zbMATH Open document ID: 1394.35253
author: Takahiro Hashira
author: Sachiko Ishida
author: Tomomi Yokota
published in: Journal of Differential Equations
publication date: 2018-03-12T00:00:00Z
review text: The authors study energy solutions to the quasilinear degenerate chemotaxis system \[ \begin{cases} u_t = \Delta u^m - \nabla \cdot (u^{q-1} \nabla v), \quad & (x,t) \in \Omega \times (0,\infty), \\ v_t = \Delta v -v +u, \quad & (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0 \in L^\infty(\Omega)\), \(v_0 \in W^{1,\infty} (\Omega)\) such that \(\nabla u_0^m \in L^2(\Omega)\). Here \(\Omega := B_R(0) \subset \mathbb{R}^N\) is a ball, \(N \geq 2\), \(m \geq 1\), and \(q \geq 2\). It is known that in case of \(q < m + \frac{2}{N}\) any energy solution is global in time and bounded, while for \(q > m + \frac{2}{N}\) there exist blow-up solutions. However, in [\textit{S. Ishida} and \textit{T. Yokota}, Discrete Contin. Dyn. Syst., Ser. B 18, No. 10, 2569--2596 (2013; Zbl 1277.35071)] it remained open whether in the latter case the blow-up occurs in finite or infinite time. The authors now answer this open question and prove that in case of \(q > m + \frac{2}{N}\) for any positive initial mass \(M\) there exists a large class of radially symmetric initial data \((u_0,v_0)\) such that \(\int_\Omega u_0 = M\) and the corresponding energy solution blows up in finite time. As the main ingredient of their proof the author show an appropriate ODI for the Lyapunov functional of the above system. This is done by an extension of the method established by \textit{M. Winkler} [J. Math. Pures Appl. (9) 100, No. 5, 748--767 (2013; Zbl 1326.35053)] and its variant from the quasilinear non-degenerate case from [\textit{T. Cieślak} and \textit{C. Stinner}, J. Differ. Equations 252, No. 10, 5832--5851 (2012; Zbl 1252.35087)] to the quasilinear degenerate case.
reviewed by: Christian Stinner
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 92C17
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 6848985
zbMATH Keywords: quasilinear degenerate Keller-Segel systems
zbMATH Keywords: finite-time blow-up
zbMATH Keywords: chemotaxis
MaRDI profile type: Publication
full work available at URL: j.jde.2018.01.038
OpenAlex ID: W2793539358
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DOI: 10.1016/J.JDE.2018.01.038
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description: scientific article; zbMATH DE number 6848985