Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source

We study the global attractors to the chemotaxis system with logistic source: ut−Δu+χ∇⋅(u∇v)=au−bu2ut−Δu+χ∇⋅(u∇v)=au−bu2, τvt−Δv=−v+uτvt−Δv=−v+u in Ω×R+Ω×R+, subject to the homogeneous Neumann boundary conditions, where smooth bounded domain Ω⊂RNΩ⊂RN, with χ,b>0χ,b>0, a∈Ra∈R, and τ∈{0,1}τ∈{0,1}. For the parabolic–elliptic case with τ=0τ=0 and N>3N>3, we obtain that the positive constant equilibrium (ab,ab) is a global attractor if a>0a>0 and b>max⁡{N−2Nχ,χa4}. Under the assumption N=3N=3, it is proved that for either the parabolic–elliptic case with τ=0τ=0, a>0a>0, b>max⁡{χ3,χa4}, or the parabolic–parabolic case with τ=1τ=1, a>0a>0, b>χa4 large enough, the system admits the positive constant equilibrium (ab,ab) as a global attractor, while the trivial equilibrium (0,0)(0,0) is a global attractor if a≤0a≤0 and b>0b>0. It is pointed out that here the convergence rates are established for all of them. The results of the paper mainly rely on parabolic regularity theory and Lyapunov functionals carefully constructed.