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Prevention of Infinite-time Blowup by Slightly Super-linear Degradation in a Keller--Segel System with Density-suppressed Motility

Yamin Xiao, Jie Jiang

Abstract

An initial-Neumann boundary value problem for a Keller--Segel system with density-suppressed motility and source terms is considered. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension $N\geq2$. In this work, we prove that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of $s\log s$ at most, the classical solution is uniformly-in-time bounded when $N\leq3$, thus preventing the infinite-time explosion detected in the source-free counter-part. The cornerstone of our proof lies in an improved comparison argument and a construction of an entropy inequality.

Prevention of Infinite-time Blowup by Slightly Super-linear Degradation in a Keller--Segel System with Density-suppressed Motility

Abstract

An initial-Neumann boundary value problem for a Keller--Segel system with density-suppressed motility and source terms is considered. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension . In this work, we prove that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of at most, the classical solution is uniformly-in-time bounded when , thus preventing the infinite-time explosion detected in the source-free counter-part. The cornerstone of our proof lies in an improved comparison argument and a construction of an entropy inequality.
Paper Structure (7 sections, 18 theorems, 142 equations)

This paper contains 7 sections, 18 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform-in-time boundedness of $v$
  4. Uniform-in-time boundedness of $u$ in 2D.
  5. Boundedness of classical solutions in 3D
  6. Hölder continuity of $v$.
  7. $L^p$-estimates for $u$.

Key Result

Theorem 1.1

Assume $N=2,3$, $\tau\geq0$ and $\gamma(s)=e^{-s}$. For any given initial data $(u^{in}, \tau v^{in})$ satisfying ini, the initial-boundary value problem ks has a unique global non-negative classical solution $(u,v)\in (C([0,\infty)\times\bar{\Omega}) \cap C^{1,2}((0,\infty)\times\bar{\Omega}))^2$,

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 27 more