Nonlocal logistics and nonlinear productions in an attraction-repulsion chemotaxis model: analysis of the global well-posedness

TL;DR

This paper analyzes a complex chemotaxis model with attraction, repulsion, and nonlocal logistic effects, proving global existence and boundedness of solutions under certain conditions, and extending previous work to include both elliptic and parabolic signal equations.

Contribution

It introduces a comprehensive analysis of a three-component chemotaxis system with nonlocal logistic terms, establishing global well-posedness for both elliptic and parabolic cases, extending prior results.

Findings

Global existence and boundedness of classical solutions

Conditions ensuring prevention of singularities

Long-time behavior analysis of solutions

Abstract

This paper investigates a {{three-component}} chemotaxis system involving both attraction and repulsion effects, as well as a nonlocal logistic-type source term. Mathematically, if $u=u(x,t)$, $v = v(x,t)$ and $w = w(x,t)$ denote the cell distribution, and the attractive and the repulsive chemical signals, the model is then described by \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \xi \nabla \cdot (u \nabla w) + a u^\alpha - b u^\alpha \int_\Omega u^\beta, & x \in \Omega, \ t > 0, \tau v_t = \Delta v - v + f(u), & x \in \Omega, \ t > 0, \tau w_t = \Delta w - w + g(u), & x \in \Omega, \ t > 0. \end{cases} \end{equation*} Here, $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) is a bounded smooth domain, $\tau\in\{0,1\}$, $a,b,\alpha,\beta,\chi,\xi>0$, the production functions $f(u)$ and $g(u)$ are assumed to satisfy algebraic growth conditions of order $\ell$ and $\rho$, generalizing prototypes of the form $u^\ell$ and $u^\rho$, $\ell,\rho>0$. The work is devoted to proving the global existence and boundedness of classical solutions under a suitable balance between the signal production exponents $\ell, \rho$ and the nonlocal damping exponents $\alpha, \beta$, for regular enough initial data and zero-flux boundary restrictions. In this regard, two main theorems are established for the cases where the chemical signals satisfy either elliptic ($\tau=0$) or parabolic ($\tau=1$) partial differential equations, highlighting how sufficiently strong nonlocal damping prevents the formation of singularities in time. We extend the results obtained in [Chiyo et al., Appl. Math. Optim. 89:9 (2024)], where the fully parabolic ($\tau=1$) and only attraction version is studied. In our context, we establish well-posedness of the system and the long-time behavior of solutions.