Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source

By Michael Winkler. 2010

803 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/a3ede7f4a305693acc5e1a3b95ee5251d75b7b97

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Citations in the corpus

Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case (2012, background): “Notice that the regularity properties of solutions to Keller-Segel type equations with reaction terms have been studied earlier in [25, 28, 27], but in a different setting.”
Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption (2022, background): “systems with two unknowns [12,20,34], while relation (11) is the so-called mass conservation property when h ≡ 0.”
Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source (2014, background): “The same conclusion is true for the parabolic–parabolic system with > 0 when either n D 2 [14, 15] or n 3 with > 0 for some 0 > 0 [16].”
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$ (2016, background): “See also [31], [44] for the study of boundedness and global existence of classical solutions when b is large.”
Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source (2015, methodology): “The effect of preventing ultimate growth has been widely studied [16, 17, 20, 19].”
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system (2015, background): “1 moreover somewhat improves the knowledge on the latter chemotaxis-only system also in the case μ > 0, for which, namely, up to now a corresponding boundedness result has been achieved only for suitably large values of μ beyond a certain number not explicitly known ([26]).”
Biomixing by Chemotaxis and Enhancement of Biological Reactions (2011, background): “ Remarks. 1. In general, chemotaxis can lead to ﬁnite time blow up of the solution (see [27] for references). However, it is known that the presence of the reaction can lead to global regularity (see [32] for a slightly diﬀerent model where references to some earlier works can also be found). We wil”
Boundedness in a two-dimensional chemotaxis-haptotaxis system (2014, background): “In the case μ = 0, solutions may blow up in finite time when n ≥ 2 [7, 13, 24]; however, it is known that arbitrarily small μ > 0 guarantee the global existence and boundedness of solutions when n = 2 [14], and that appropriately large μ χ preclude blow-up in the case n ≥ 3 [22].”
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening (2014, background): “Thus going beyond the boundedness results in [23], [22] and [32], for suitably large values of μ χ this entirely rules out any type of persisting oscillatory behavior, such as detected numerically in [12] for appropriate choices of χ and μ.”
Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source (2015, methodology): “Here, we note that, none of these results except [50] can be applied to the simplest minimal-chemotaxis-growth model (1.”
Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals (2015, background): “various mechanisms have been identified to prevent finite-time blow-up, such as considering nonlinear variants of chemotactic sensitivity and diffusivity ([30], [4], [35]), incorporating logistic dampening ([27], [38]), or also adding a cross-diffusion term in the equation for the chemical signal ([”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, methodology): “3) is well-established by using a suitable fixed point argument and standard parabolic regularity theory; see, for example, [20, 29, 43, 53].”
Global classical solution and boundedness to a chemotaxis‐haptotaxis model with re‐establishment mechanisms (2018, background): “When μ > 0, the solution will always exist globally for any χ, μ > 0 when N = 2 [11], but for the three-dimensional space, the global solution only be established for large μ χ [23], and this problem is left open for small μ χ .”