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

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Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion (2019, methodology): “On the other hand, for κ = 1, He and Zheng [6] modified the energy functional method from [1] to obtain the stabilities of the constant equilibria (0, 0) and (a/b, a/b) for κ = 1 with convergence rate estimates.”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, methodology): “To this end, we study the large time behavior for the minimal chemotaxis model with a standard logistic source: ut = ∇ · (d1∇u− χu∇v) + κu− µu2, x ∈ Ω, t > 0, vt = d2∆v − βv + αu, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x) ≥ 0, v(x, 0) = v0(x) ≥ 0, x ∈ Ω. (1.8) For d1”
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016, background): “2) uniformly converge to the constant equilibrium ([2 4]); convergence rates have been considered in [3].”
Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation (2018, background): “Indeed, the analysis on qualitative aspects of solution behavior in chemotaxis systems with logistic kinetics seems widely dominated by studies concerned with effects already known from the corresponding taxis-free and hence diagnonal reaction-diffusion systems; accordingly available findings on asy”