Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

Zheng, Pan; Mu, Chunlai; Willie, Robert; Hu, Xuegang

doi:10.1016/j.camwa.2017.11.032

Cited by 20 publications

(7 citation statements)

References 21 publications

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“…When the system (1.2) has a logistic source f (u) and g ≡ 0, θ ≡ 1, as f (u) = ru − µu 2 , Zhao and Zheng [54] proved that the system (1.2) possesses a global bounded classical solutions if r > χ 2 4 for 0 < χ ≤ 2, or r > χ−1 for χ > 2 in two-dimensional bounded domains. Based on existent results in [54], Zheng et al [58] showed that the global bounded solution (u, v) exponentially converges to the steady state ( r µ , r µ ). Zhao and Zheng [56] generalized their own work [54] to the higher dimensional case.…”

Section: When the Second Equation Degenerates Into An Elliptic Equation ∇Vmentioning

confidence: 99%

Global existence in a chemotaxis system with singular sensitivity and signal production

Ren¹,

Ma²

2022

DCDS-B

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Section: When the Second Equation Degenerates Into An Elliptic Equation ∇Vmentioning

confidence: 99%

Global existence in a chemotaxis system with singular sensitivity and signal production

Ren¹,

Ma²

2022

DCDS-B

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“…In this section, inspired by [1, 9, 15, 20, 21, 27], we prove the result in Theorem 1.2 by establishing an appropriate Lyapunov function. We shall establish asymptotic stability of global solutions to (1.2) based upon the regularities of global bounded solutions.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Large time behavior of solutions to a chemotaxis system with singular sensitivity and logistic source

2021

Mathematische Nachrichten

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In this paper, we study the following chemotaxis system with singular sensitivity and logistic source trueright85.0pt{ut=Δu−χ∇·0trueuv∇v+ru−μuk,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,in a smooth bounded convex domain Ω⊂double-struckRn (n≥2) with the non‐flux boundary, where χ,r,μ>0,k>2. The boundedness of solutions has been proved in the case that n≥2, k>3(n+2)n+4 and r, χ>0 satisfying χ2

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“…In view of the global attractivity discussed above, we only need to analyze its local stability. By applying some abstract stability results based on analytic semigroup theories (see [5,19]), to get the stability of the spatially homogeneous steady state (u * , v * , w * , z * ), it suffices to prove that the steady state is spectrally stable, i.e., the linearized operator has only eigenvalues with nonnegative real parts (also see [4,31,38]). Linearizing (3) at (u * , v * , w * , z * ) leads to the following system…”

Section: Huanhuan Qiu and Shangjiang Guomentioning

confidence: 99%

Global existence and stability in a two-species chemotaxis system

Qiu¹,

Guo²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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This paper deals with the following two-species chemotaxis system        ut = ∆u − χ 1 ∇ • (u∇v) + µ 1 u(1 − u − a 1 w), x ∈ Ω, t > 0, vt = ∆v − v + h(w), x ∈ Ω, t > 0, wt = ∆w − χ 2 ∇ • (w∇z) + µ 2 w(1 − w − a 2 u), x ∈ Ω, t > 0, zt = ∆z − z + h(u), x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C 1-regular function. The main objectives of this paper are twofold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium (u * , v * , w * , z *) may be globally attractive in the weak competition case (i.e., 0 < a 1 , a 2 < 1), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., a 1 > 1 > a 2 > 0). In the fully strong competition case (i.e. a 1 , a 2 > 1), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous (u * , v * , w * , z *). The matter which species ultimately wins out depends crucially on the starting advantage each species has.

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Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

Cited by 20 publications

References 21 publications

Global existence in a chemotaxis system with singular sensitivity and signal production

Global existence in a chemotaxis system with singular sensitivity and signal production

Large time behavior of solutions to a chemotaxis system with singular sensitivity and logistic source

Global existence and stability in a two-species chemotaxis system

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