Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

Jing Zhang,Li Qu,Liangchen Wang,Xuegang Hu

doi:10.14232/ejqtde.2019.1.31

Abstract

Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

Highlights

This paper considers the following quasilinear chemotaxis system ut = ∇ · (D1(u)∇u) − ∇ · (uχ1(w)∇w) + μ1u(1 − u − a1v), vt = ∇ · (D2(v)∇v) − ∇ · (vχ2(w)∇w) + μ2v(1 − a2u − v)
Motivated by the arguments in [19,29,30,37,41], in this paper, we extend their method and obtain global boundedness of solution of model (1.1)
Theorem 1.1 gives a qualitative result, namely, if μi (i = 1, 2) are sufficiently large, model (1.1) has global bounded solutions, which improves above results in some sense

Summary

Introduction

D(u) denotes the diffusion function and f (u) is the logistic source The analysis of this model has attracted many interests and many results are presented. When D(u) satisfies (1.4), Wang et al prove that system (1.5) possesses a unique global bounded classical solution if m. If the logistic source f (u) = au − μuγ with γ > 1 and D(u) = δ in system (1.5), the global bounded solution is studied if. The chemotaxis-consumption model (1.5) with nonlinear diffusion function and nontrivial source terms has already been considered in [28, 30]. Theorem 1.1 gives a qualitative result, namely, if μi (i = 1, 2) are sufficiently large, model (1.1) has global bounded solutions, which improves above results in some sense.

Preliminaries

Global boundedness