Boundedness of Classical Solutions to a Degenerate Keller–Segel Type Model with Signal-Dependent Motilities

Kentaro Fujie, Jie Jiang

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we consider the initial Neumann boundary value problem for a degenerate kinetic model of Keller–Segel type. The system features a signal-dependent decreasing motility function that vanishes asymptotically, i.e., degeneracies may take place as the concentration of signals tends to infinity. In the present work, we are interested in the boundedness of classical solutions when the motility function satisfies certain decay rate assumptions. Roughly speaking, in the two-dimensional setting, we prove that classical solution is globally bounded if the motility function decreases slower than an exponential speed at high signal concentrations. In higher dimensions, boundedness is obtained when the motility decreases at certain algebraical speed. The proof is based on the comparison methods developed in our previous work (Fujie and Jiang in J. Differ. Equ. 269:5338–5778, 2020; Fujie and Jiang in Calc. Var. Partial Differ. Equ. 60:92, 2021) together with a modified Alikakos–Moser type iteration. Besides, new estimations involving certain weighted energies are also constructed to establish the boundedness.

Original languageEnglish
Article number3
JournalActa Applicandae Mathematicae
Volume176
Issue number1
DOIs
Publication statusPublished - 2021 Dec

Keywords

  • Boundedness
  • Chemotaxis
  • Classical solutions
  • Degeneracy
  • Keller–Segel models

ASJC Scopus subject areas

  • Applied Mathematics

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