On a fully parabolic chemotaxis system with Lotka–Volterra competitive kinetics

Abstract

This paper is devoted to the following fully parabolic chemotaxis system with Lotka–Volterra competitive kinetics { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v − χ 2 ∇ ⋅ ( v ∇ w ) + μ 2 v ( 1 − v − a 2 u ) , x ∈ Ω , t > 0 , w t = Δ w − λ w + b 1 u + b 2 v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions, where Ω ⊂ R n is a bounded domain with smooth boundary. We mainly consider the global existence and boundedness of classical solutions in the three dimensional case, which extends and partially improves the results of Bai–Winkler (2016) [1] , Xiang (2018) [25] , as well as Lin–Mu–Wang (2015) [10] , etc.