Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system

Winkler, Michael

doi:10.1016/j.matpur.2013.01.020

Cited by 809 publications

(452 citation statements)

References 29 publications

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“…Anyhow, a highly destabilizing potential of cross-diffusive terms of the type in (1.3), at relative strength increasing with the spatial dimension, is indicated by known results on the related classical Keller-Segel system of chemotaxis, as obtained by replacing the second equation in (1.3) with c t = ∆c − c + n: While all classical solutions to the corresponding initial-boundary value problem remain bounded when either N = 1, or N = 2 and the total mass Ω n 0 of cells is small ( [24], [23]), it is known that finite-time blow-up does occur for large classes of radially symmetric initial data when either N = 2 and Ω n 0 is large, or N ≥ 3 and Ω n 0 is an arbitrarily small prescribed number ( [22], [38]). …”

Section: Introductionmentioning

confidence: 99%

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

Winkler

2013

Arch Rational Mech Anal

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385

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The chemotaxis-Navier-Stokes system          n t + u · ∇n = ∆n − ∇ · (nχ(c)∇c), c t + u · ∇c = ∆c − nf (c),is considered under homogeneous boundary conditions of Neumann type for n and c, and of Dirichlet type for u, in a bounded convex domain Ω ⊂ R 3 with smooth boundary, where Φ ∈ W 1,∞ (Ω), and where f ∈ C 1 ([0, ∞)) and χ ∈ C 2 ([0, ∞)) are nonnegative with f (0) = 0. Problems of this type have been used to describe the mutual interaction of populations of swimming aerobic bacteria with the surrounding fluid. Up to now, however, global existence results seem to be available only for certain simplified variants such as e.g. the two-dimensional analogue of (⋆), or the associated chemotaxis-Stokes system obtained on neglecting the nonlinear convective term in the fluid equation.The present work gives an affirmative answer to the question of global solvability for (⋆) in the following sense: Under mild assumptions on the initial data, and under modest structural assumptions on f and χ, inter alia allowing for the prototypical case when f (s) = s for all s ≥ 0 and χ ≡ const., the corresponding initial-boundary value problem is shown to possess a globally defined weak solution.This solution is obtained as the limit of smooth solutions to suitably regularized problems, where appropriate compactness properties are derived on the basis of a priori estimates gained from an energy-type inequality for (⋆) which in an apparently novel manner combines the standard L 2 dissipation property of the fluid evolution with a quasi-dissipative structure associated with the chemotaxis subsystem in (⋆).

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Section: Introductionmentioning

confidence: 99%

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

Winkler

2013

Arch Rational Mech Anal

Self Cite

385

162

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“…For γ 0 = 1, n = 3, Winkler [10], [11] prove that for any prescribed total mass of cells, there exist radially symmetric positive initial data such that the corresponding solution blows up in finite time. Dynamics and sensitivity of the solutions for a bacterial self-organization model is investigated in [4].…”

Section: Numerical Tests and Conclusionmentioning

confidence: 99%

“…Dynamics and sensitivity of the solutions for a bacterial self-organization model is investigated in [4]. Since the existence of cell kinetics term f (u, w) and without the assumption of radially symmetric in this paper, we can't construct effective energy function as done in [5], [11]. Our future research is to study the blow-up issue of the system (1.1).…”

Section: Numerical Tests and Conclusionmentioning

confidence: 99%

Global Existence and Blow-Up for a Chemotaxis System

Chen

Zhang

et al. 2017

Mathematical Modelling and Analysis

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Abstract. In this paper we consider a Keller-Segel-type chemotaxis model with reaction term under no-flux boundary conditions, where the kinetics term of the system is power function. Assuming some growth conditions, the existence of bounded global strong solution to the parabolic-parabolic system is given. We also give the numerical test and find out that there exists a threshold. When the power frequency greater than the threshold, both global solution and blow-up solution exist.

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“…In 1970s, a well-known chemotaxis model was proposed by Keller and Segel ([13]), which describes the aggregation processes of the cellular slime mold Dictyostelium discoideum. Since then, a number of variations of the Keller-Segel model have attracted the attention of many mathematicians, and the focused issue was the boundedness or blow-up of the solutions ( [5,7,9,10,39,20]). The striking feature of Keller-Segel models is the possibility of blow-up of solutions in a finite (or infinite) time (see, e.g., [1,9,18,39]), which strongly depends on the space dimension.…”

Section: Introductionmentioning

confidence: 99%

“…Since then, a number of variations of the Keller-Segel model have attracted the attention of many mathematicians, and the focused issue was the boundedness or blow-up of the solutions ( [5,7,9,10,39,20]). The striking feature of Keller-Segel models is the possibility of blow-up of solutions in a finite (or infinite) time (see, e.g., [1,9,18,39]), which strongly depends on the space dimension. Moreover, some recent studies have shown that the blow-up of solutions can be inhibited by the nonlinear diffusion (see Ishida et al [11] Winkler et al [1,27,36,40]) and the (generalized) logistic damping (see Li and Xiang [14], Tello and Winkler [31], Wang et al [33], Zheng et al [48]).…”

Section: Introductionmentioning

confidence: 99%