Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

By Johannes Lankeit. 2014

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Abstract We prove existence of global weak solutions to the chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ u − μ u 2 v t = Δ v − v + u under homogeneous Neumann boundary conditions in a smooth bounded convex domain Ω ⊂ R n , for arbitrarily small values of μ > 0 . Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that κ is not too large. In this case, we also consider their large-time behaviour: We prove decay if κ ≤ 0 and the existence of an absorbing set if κ > 0 is sufficiently small.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/816732456a73076380bffbe530263e69f2e57a7a

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Citations in the corpus

Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation (2018, background): “In the latter context, for κ = 2 and arbitrary n ≥ 3 it could more recently be shown in [17] that any μ > 0 is sufficient to warrant global solvability at least in the natural framework of weak solutions which in the case n = 3 moreover, though possibly undergoing some explosions within finite time,”
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems (2017, background): “the additional dissipative effect of the quadratic zero-order death term is known to rule out any such collapse when either n = 2 and μ > 0 is arbitrary ([17]), or n ≥ 3 and μ is sufficiently large ([25]); if in the latter case n ≥ 3 the number μ > 0 is arbitrary, then at least certain global weak s”
Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source (2016, background): “2 of [12] ensures that actually T = ∞; nally, since β > n+ 2, W (Ω) ⊂ L∞(Ω), so that (uε, vε) is a global classical solution on Ω× (0,∞) to problem (4).”
A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “2) in cases when μ is positive but small, in which only weak solutions are known to exist globally ([35]).”
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016, background): “For small μ > 0, at least global weak solutions are known to exist ([7]), and in 3-dimensional domains and for small κ, their large-time behaviour has been investigated ([7]).”
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$ (2016, background): “When b is small, among others, Lankeit in [23] proved the existence of at least one global weak solution with given initial functions.”
On the weakly competitive case in a two-species chemotaxis model (2016, background): “1) (studied in [42, 49, 50, 22]), gives rise to colorful dynamics as witnessed in e.”