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Global dynamics of chemotaxis-consumption systems with oppositely acting nonlocal terms

Rafael Diaz Fuentes, Fatma Gamze Duzgun, Silvia Frassu, Giuseppe Viglialoro

TL;DR

The paper addresses global existence and uniform boundedness for a Keller–Segel chemotaxis–consumption system with nonlocal logistic terms in a bounded domain. It analyzes two nonlocal logistic forms, one with nonlocal damping and the other with nonlocal growth, and derives a boundedness criterion together with detailed a priori estimates. Using maximal regularity tools, Sobolev embeddings, and Gagliardo–Nirenberg inequalities, the authors identify explicit exponent regimes that guarantee global classical solutions with $u,v$ bounded for all times; special care is given to the mass behavior in the first form of $f(u)$. The results extend the literature on KS systems by incorporating nonlocal logistic feedback in a consumption framework and clarify how different nonlocal reaction terms shape long-time dynamics, including the absence of a logistic term recovering known consumption models.

Abstract

This paper studies a chemotaxis system where cells move in response to a chemical signal within a confined habitat. The model includes external source terms that combine local and nonlocal growth with dampening effects. The main focus is on conditions under which solutions exist for all time and remain uniformly bounded, preventing cell aggregation. Two types of source terms are considered. In the first case, the structure of the source term ensures that the total cell mass remains controlled over time. In the second case, this mass control is not guaranteed, which can lead to different dynamic behaviors. The results extend previous studies that examined similar systems but with more specific source terms and slightly different chemical dynamics. This work highlights how variations in the reaction terms influence the long-term behavior of the system.

Global dynamics of chemotaxis-consumption systems with oppositely acting nonlocal terms

TL;DR

The paper addresses global existence and uniform boundedness for a Keller–Segel chemotaxis–consumption system with nonlocal logistic terms in a bounded domain. It analyzes two nonlocal logistic forms, one with nonlocal damping and the other with nonlocal growth, and derives a boundedness criterion together with detailed a priori estimates. Using maximal regularity tools, Sobolev embeddings, and Gagliardo–Nirenberg inequalities, the authors identify explicit exponent regimes that guarantee global classical solutions with bounded for all times; special care is given to the mass behavior in the first form of . The results extend the literature on KS systems by incorporating nonlocal logistic feedback in a consumption framework and clarify how different nonlocal reaction terms shape long-time dynamics, including the absence of a logistic term recovering known consumption models.

Abstract

This paper studies a chemotaxis system where cells move in response to a chemical signal within a confined habitat. The model includes external source terms that combine local and nonlocal growth with dampening effects. The main focus is on conditions under which solutions exist for all time and remain uniformly bounded, preventing cell aggregation. Two types of source terms are considered. In the first case, the structure of the source term ensures that the total cell mass remains controlled over time. In the second case, this mass control is not guaranteed, which can lead to different dynamic behaviors. The results extend previous studies that examined similar systems but with more specific source terms and slightly different chemical dynamics. This work highlights how variations in the reaction terms influence the long-term behavior of the system.

Paper Structure

This paper contains 17 sections, 11 theorems, 75 equations.

Table of Contents

  1. Introduction and selected known results
  2. An overview on Keller--Segel models
  3. An overview of Keller--Segel models with production and logistic effects
  4. Classical logistics
  5. Nonlocal logistics
  6. Gradient-dependent logistics
  7. Presentation of the model and motivations; claim of the main result
  8. Basic description of the model
  9. Interpretation of $f(u)$ and its role in the investigation
  10. Formulation of the main Theorem
  11. Preliminary results
  12. Local existence, parabolic regularity results and globality
  13. A boundedness criterion
  14. A priori estimates
  15. The case $f(u)=a u^\alpha-bu^\beta \int_\Omega u^\gamma$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let the hypotheses in reglocal be fulfilled and $\chi, b>0$ and $\alpha,\gamma \geq 1$. Then problem problem admits a unique solution with $0\leq u,v \in L^\infty(\Omega \times (0,\infty))$, in these circumstances:

Theorems & Definitions (25)

  • Remark 1: Consumption models with classical and gradient-dependent logistics
  • Theorem 2.1
  • Remark 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 15 more