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Suppression of Blowup by Slightly Superlinear Degradation in a Parabolic-Elliptic Keller--Segel System with Signal-dependent Motility

Aijing Lu, Jie Jiang

Abstract

In this paper, we consider an initial-Neumann boundary value problem for a parabolic-elliptic Keller-Segel system with signal-dependent motility and a source term. Previous research has rigorously shown that the source-free version of this system exhibits an infinite-time blowup phenomenon when dimension $N \geq 2$. In the current work, when $N \leq 3$, we establish uniform boundedness of global classical solutions with an additional source term that involves slightly super-linear degradation effect on the density, of a maximum growth order $s\log s$, unveiling a sufficient blowup suppression mechanism. The motility function considered in our work takes a rather general form compared with recent works \cite{FuJi2020, LyWa2023} which were restricted to the monotone non-increasing case. The cornerstone of our proof lies in deriving an upper bound for the second component of the system and an entropy-like estimate, which are achieved through tricky comparison skills and energy methods, respectively.

Suppression of Blowup by Slightly Superlinear Degradation in a Parabolic-Elliptic Keller--Segel System with Signal-dependent Motility

Abstract

In this paper, we consider an initial-Neumann boundary value problem for a parabolic-elliptic Keller-Segel system with signal-dependent motility and a source term. Previous research has rigorously shown that the source-free version of this system exhibits an infinite-time blowup phenomenon when dimension . In the current work, when , we establish uniform boundedness of global classical solutions with an additional source term that involves slightly super-linear degradation effect on the density, of a maximum growth order , unveiling a sufficient blowup suppression mechanism. The motility function considered in our work takes a rather general form compared with recent works \cite{FuJi2020, LyWa2023} which were restricted to the monotone non-increasing case. The cornerstone of our proof lies in deriving an upper bound for the second component of the system and an entropy-like estimate, which are achieved through tricky comparison skills and energy methods, respectively.
Paper Structure (12 sections, 23 theorems, 148 equations)

This paper contains 12 sections, 23 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-independent upper bounds for $v$
  4. Uniform upper bound for $v$ with bounded motility
  5. Uniform upper bound for $v$ with unbounded motility
  6. Time-independent upper bounds for $u$ in 2D
  7. Time-independent upper bounds for $u$ in 3D
  8. Uniform in-time entropy estimate
  9. Hölder Continuity of $v$
  10. Higher-order estimates of $v$
  11. Uniform-in-time boundedness for $\|u\|_{L^q(\Omega)}$ with $q>3/2$
  12. Proof of Theorem \ref{['thm2']}

Key Result

theorem 1

Assume $\Omega \subset \mathbb{R}^N$ with $N\leq3$. Suppose that $f(\cdot)$ satisfies the conditions f1-f2, and that $\gamma(\cdot)$ satisfies gamma. For any given initial datum $u^{in}$ satisfying uin, problem 0.1 has a unique global non-negative classical solution $(u,v)\in \left( C^0([0,\infty)\t

Theorems & Definitions (43)

  • theorem 1
  • theorem 2
  • theorem 3
  • lemma 1
  • lemma 2
  • lemma 3
  • proof
  • remark 1
  • lemma 4
  • proposition 1
  • ...and 33 more