Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform boundedness for the two-dimensional Keller-Segel system with Gompertz growth

Nohayla Alaoui, Mohamed Halloumi, Giuseppe Viglialoro

Abstract

It is known that in two dimensions the classical Keller-Segel model can lead to cell aggregation. This behavior can be controlled by adding a logistic growth term with quadratic decay. Researchers have tried to find weaker damping mechanisms that can still stabilize the system. Previous work showed that, under suitable assumptions on the initial cell distribution, even weaker growth terms than the classical logistic one can prevent aggregation. In this paper, we study the effect of a Gompertz-type growth term in a minimal two-dimensional chemotaxis model. This term provides a weaker damping effect than those previously considered. We analyze how it influences the system and identify conditions that guarantee that solutions exist for all time and remain bounded.

Uniform boundedness for the two-dimensional Keller-Segel system with Gompertz growth

Abstract

It is known that in two dimensions the classical Keller-Segel model can lead to cell aggregation. This behavior can be controlled by adding a logistic growth term with quadratic decay. Researchers have tried to find weaker damping mechanisms that can still stabilize the system. Previous work showed that, under suitable assumptions on the initial cell distribution, even weaker growth terms than the classical logistic one can prevent aggregation. In this paper, we study the effect of a Gompertz-type growth term in a minimal two-dimensional chemotaxis model. This term provides a weaker damping effect than those previously considered. We analyze how it influences the system and identify conditions that guarantee that solutions exist for all time and remain bounded.
Paper Structure (4 sections, 4 theorems, 35 equations)

This paper contains 4 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction, motivation and presentation of the main result
  2. Local existence and boundedness criterion
  3. A priori estimates and proof of Thorem \ref{['MainTheorem']}
  4. Proof of Theorem \ref{['MainTheorem']}

Key Result

Theorem 1.1

Under assumptions AssumptionDomain, there exists a positive constant $C_{GN}$ such that whenever being $M:=\max\left\{\int_\Omega u_0(x)dx,K|\Omega|\right\}$, it is possible to find a uniquely determined pair of functions $(u,v) \in C^{2+\delta,\,1+\frac{\delta}{2}}(\overline{\Omega} \times [0,\infty)) \times C^{2+\delta,\,\tau+\frac{\delta}{2}}(\overline{\Omega} \times [0,\infty)),$ solving pr

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof