Global solutions in a high-dimensional two-species chemotaxis model with Lotka–Volterra competitive kinetics

Abstract

This paper deals with the two-species chemotaxis system with logistic source{ut=Δu−χ1∇⋅(u∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=Δv−χ2∇⋅(v∇w)+μ2v(1−a2u−v),x∈Ω,t>0,wt=Δw−λw+αu+βv,x∈Ω,t>0 under homogeneous Neumann boundary condition in a smooth bounded domain Ω⊂Rn(n≥1). It is proved that in convex domains the problem possesses a unique global bounded solution if μ1 and μ2 are large enough. Moreover, we establish the existence of global weak solution for any μ1>0 and μ2>0.