Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant

We consider chemotaxis system with consumption of chemoattractant {vt=Δv−uv,ut=Δu−χ∇⋅(u∇v), under homogeneous Neumann boundary conditions. It is proved that if either n ≤ 2 or 0<χ≤16(n+1)‖v(x,0)‖L∞(Ω), n ≥ 3, the global classical solution (u, v) of this problem converges to (ū0,0) exponentially as t → ∞, where ū0≔1|Ω|∫Ωu(x,0)dx.