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

Abstract

Please cite this article in press as: X. He, S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl. (2016), http://dx. AbstractWe study the global attractors to the chemotaxis system with logistic source:and τ ∈ {0, 1}. For the parabolic-elliptic case with τ = 0 and N > 3, we obtain that the positive constant equilibrium ( a b , a b ) is a global attractor if a > 0 and b > max{ N −2 N χ, χ √ a 4 }. Under the assumption N = 3, it is proved that for either the parabolic-elliptic case with τ = 0, a > 0, b > max{ χ 3 , χ √ a 4 }, or the parabolic-parabolic case with τ = 1, a > 0, b > χ √ a 4 large enough, the system admits the positive constant equilibrium ( a b , a b ) as a global attractor, while the trivial equilibrium (0, 0) is a global attractor if a ≤ 0 and b > 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. 2010MSC: 35K55, 35B40, 35Q92, 92C17

Global Existence and Asymptotic Behavior of Classical Solutions for a 3D Two-Species Keller-Segel-Stokes System with Competitive Kinetics

“…In the higher dimensional case, Winkler [27,29] established existence of global classical solutions under the condition that µ > 0 is sufficiently large. Moreover, asymptotic behavior of the solutions was obtained: n(·, t) → r µ , c(·, t) → r µ in L ∞ (Ω) as t → ∞ ( [7]). Recently, Lankeit [13] obtained global existence of weak solutions to the chemotaxis system with logistic source for all r ∈ R and any µ > 0 and the eventual smoothness of the solutions was derived if r ∈ R is small.…”

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

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How strongly does diffusion or logistic-type degradation affect existence of global weak solutions in a chemotaxis–Navier–Stokes system?

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