On boundedness, gradient estimate, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop

TL;DR

This paper analyzes the conditions for boundedness, blow-up, and convergence of solutions in a two-species chemotaxis system, revealing how the product of species masses influences these behaviors in a 2D domain.

Contribution

It identifies a mass product criterion for global boundedness and convergence, and characterizes blow-up conditions in a chemotaxis model with/without signaling loops.

Findings

Smallness of the product of species masses ensures boundedness and convergence.

Large product of masses can lead to blow-up along a specific line.

The results extend understanding of chemotaxis dynamics in 2D domains.

Abstract

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,\infty}(1\leq j\leq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and $W^{j,\infty}$-convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model.