Dynamic behavior analysis to a generalized chemotaxis-consumption system

Abstract

In this paper, the following chemotaxis system with nonlinear consumption mechanisms is considered ut = Δu − χ∇ · (u∇v) + ξ∇ · (u∇w) + au − bum, vt = Δv − uαv, wt = Δw − uβw under homogeneous Neumann boundary conditions, where Ω⊂Rn(n≥2) is a smoothly bounded domain and parameters χ, ξ, a, b, α, β > 0 and m > 1. If m and l = max{α, β} satisfy m>maxl(n+2)2,1, then the system possesses a global classical solution, which is bounded in Ω × (0, ∞). Furthermore, it has been shown that such solution exponentially converges to equilibrium ((ab)1m−1,0,0) as t → ∞, where convergence rate can be formally characterized by the parameters of the system.