Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a Keller--Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources

Yujiao Sun, Jie Jiang

Abstract

This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function $γ(v) = e^{-v}$, a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.

On a Keller--Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources

Abstract

This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function , a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.

Paper Structure

This paper contains 15 sections, 31 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Introduction of auxiliary functions
  4. Global existence for homogeneous case
  5. Upper bound for $v$
  6. Upper bound for $u$
  7. Uniform-in-time boundedness for homogeneous case
  8. Time-independent upper bound for $v$
  9. Time-independent upper bound for $u$
  10. The critical mass phenomenon for $\gamma(v)=e^{-v}$ in 2D
  11. Uniform-in-time boundedness with sub-critical mass
  12. Unboundedness with super-critical mass
  13. Globally boundedness for nonhomogeneous case
  14. Uniform-in-time upper bound for $v$
  15. Uniform-in-time upper bound for $u$

Key Result

theorem 1

Let $\Omega$ be a bounded domain of $\mathbb{R}^N$($N\geq1$) with smooth boundary. Suppose that the initial data $(u_0,v_0,h_0)$ satisfies condition A0, the motility function $\gamma(\cdot)$ satisfies assumption A1 and $f \equiv 0$. Then problem Our Problem has a unique global nonnegative classical $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (55)

  • theorem 1
  • remark 1
  • theorem 2
  • theorem 3
  • theorem 4
  • remark 2
  • theorem 5
  • proof
  • lemma 1
  • proof
  • ...and 45 more