Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics

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Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions (2020, background): “10) Moreover, for such α we see that φ2(ξ) := ξ α−1−1− (α−1) ·2α−2(ξ−1), ξ ∈ [1, 2], has the properties that φ2(1) = 0 and φ ′ 2(ξ) = (α−1) ·ξα−2−(α−1) ·2α−2 ≥ 0 for all ξ ∈ (1, 2), whence φ2 is nonnegative on [1, 2].”
Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation (2018, background): “1), inter alia accounting for nonlinear cell diffusion ([51], [6], [41], [55], [58]), for variants in the cross-diffusive interaction ([53], [20], [4], [39], [57]) or also in the signal evolution, or even containing couplings to further mechanisms and components such as haptotactic cues ([5], [40], ”
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.”