Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

Mizukami, Masaaki

doi:10.1002/mma.4607

Cited by 49 publications

(19 citation statements)

References 34 publications

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“…After the above pioneering work, the Keller-Segel system was studied intensively; conditions for global existence or blow-up in the parabolic-elliptic system were studied in [24,27] and in the parabolic-parabolic system were investigated in [2,5,[12][13][14]27,35]; blow-up asymptotics of solutions for the parabolic-elliptic system is in [29,30] and for the parabolic-parabolic system is in [13,19,26]. More related works can be found in [1,4,18,20,21,[31][32][33]36]; a chemotaxis system with logistic term in the parabolic-elliptic case is in [32] and in the parabolicparabolic case is in [36]; global existence and stabilization in a two-species chemotaxis-competition system were shown in the parabolic-parabolic-elliptic case ( [4,21,31,33]) and in the parabolic-parabolic-parabolic case ( [1,18,20]).…”

Section: Introductionmentioning

confidence: 99%

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

Mizukami

2018

Mathematische Nachrichten

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Abstract. This paper gives a first insight into making a mathematical bridge between the parabolic-parabolic signal-dependent chemotaxis system and its parabolic-elliptic version.To be more precise, this paper deals with convergence of a solution for the parabolicparabolic chemotaxis system with strong signal sensitivityto that for the parabolic-elliptic chemotaxis systemwhere Ω is a bounded domain in R n (n ∈ N) with smooth boundary, λ > 0 is a constant and χ is a function generalizingIn chemotaxis systems parabolic-elliptic systems often provided some guide to methods and results for parabolic-parabolic systems. However, the relation between parabolicelliptic systems and parabolic-parabolic systems has not been studied. Namely, it still remains to analyze on the following question: Does a solution of the parabolic-parabolic system converge to that of the parabolic-elliptic system as λ ց 0? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.2010Mathematics Subject Classification. Primary: 35A09; Secondary: 35K51, 35J15, 92C17.

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Section: Introductionmentioning

confidence: 99%

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

Mizukami

2018

Mathematische Nachrichten

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“…in L ∞ (Ω) as t → ∞ in the case that a 1 , a 2 ∈ (0, 1), and n 1 (·, t) → 0, n 2 (·, t) → 1, c(·, t) → β in L ∞ (Ω) as t → ∞ in the case that a 1 ≥ 1 > a 2 > 0 ( [1,16,17]). More related works can be found in [3,15,18,22,25].…”

Section: Introduction and Resultsmentioning

confidence: 99%

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

Cao

Kurima

Mizukami³

2019

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This paper deals with the two-species Keller-Segel-Stokes system with competitive kinetics                (n 1 ) t + u · ∇n 1 = ∆n 1 − χ 1 ∇ · (n 1 ∇c) + µ 1 n 1 (1 − n 1 − a 1 n 2 ), x ∈ Ω, t > 0, (n 2 ) t + u · ∇n 2 = ∆n 2 − χ 2 ∇ · (n 2 ∇c) + µ 2 n 2 (1 − a 2 n 1 − n 2 ), x ∈ Ω, t > 0,under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 3 with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where −c + αn 1 + βn 2 is replaced with −(αn 1 + βn 2 )c in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that µ 1 , µ 2 are sufficiently large ([5]). Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the L ∞ -information of c. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for µ 1 , µ 2 .2010Mathematics Subject Classification. Primary: 35K45; Secondary: 92C17; 35Q35.

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“…Recently, the conditions for asymptotic behavior in the case of a 1 , a 2 ∈ (0, 1) were once more improved by Mizukami [20]. Moreover, the global solution and large time behavior have also been established for the parabolic-parabolic-elliptic chemotaxis model [3,17,30,35].…”

Section: 2)mentioning

confidence: 99%

Boundedness and stabilization in a two-species chemotaxis system with two chemicals

Wang¹,

Zhang²,

Mu³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - B

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This paper deals with the two-species chemotaxis system with two chemicalsx ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n (n ≥ 1), where the parameters d 1 , d 2 , d 3 , d 4 > 0, µ 1 , µ 2 > 0, a 1 , a 2 > 0 and α, β > 0. The chemotactic function χ i (i = 1, 2) and the signal production function f i (i = 1, 2) are smooth. If n = 2, it is shown that this system possesses a unique global bounded classical solution provided that |χ i | (i = 1, 2) are bounded. If n ≤ 3, this system possesses a unique global bounded classical solution provided that µ i (i = 1, 2) are sufficiently large. Specifically, we first obtain an explicit formula µ i0 > 0 such that this system has no blow-up whenever µ i > µ i0 . Moreover, by constructing suitable energy functions, it is shown that:• If a 1 , a 2 ∈ (0, 1) and µ 1 and µ 2 are sufficiently large, then any global bounded solution exponentially converges to• If a 1 > 1 > a 2 > 0 and µ 2 is sufficiently large, then any global bounded solution exponentially converges to (0, f 1 (1)/α, 1, 0) as t → ∞;• If a 1 = 1 > a 2 > 0 and µ 2 is sufficiently large, then any global bounded solution algebraically converges to (0, f 1 (1)/α, 1, 0) as t → ∞. 2010 Mathematics Subject Classification. Primary: 35K35, 35A01, 35B44; Secondary: 35B35, 92C17.

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Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

Cited by 49 publications

References 34 publications

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

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

Boundedness and stabilization in a two-species chemotaxis system with two chemicals

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