Global boundedness of solutions to a two-species chemotaxis system

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On the weakly competitive case in a two-species chemotaxis model (2016, background): “In [54], for different sensitivity functions satisfying χi(w) ≤ Ki (1+αiw)i for some ki > 1, i ∈ {1, 2}, global bounded classical solutions were proven to exist under the condition of χ, μ,K being sufficiently small (where the meaning of ’sufficiently small’ depends on the initial mass).”
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], ”