An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model

Zhang, Yajing; Chen, Xinfu; Hao, Jianghao; Lai, Xingping; Qin, Cong

doi:10.1016/j.jmaa.2014.06.005

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…Unlike the boundary spike which is shown stable in [4,47,49], we find that both the double-boundary spike and the single interior spike are unstable. A natural extension is to analyze the stability of similar multi-spike solutions.…”

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confidence: 64%

“…To further expose the dynamics of the single boundary spike established by [47], X. Chen et al [4,49] obtain its refined asymptotic expansion and prove its locally and exponentially asymptotical stability. To be precise, they find that the single boundary spike of (2.1) is asymptotically given by…”

mentioning

confidence: 99%

“…We now prove that the eigenvalue λ d is real, and we follow the novel approach in [49] with few necessary modifications. Firstly, we multiply the first equation of (2.3) by the conjugate of w d /κ and integrate it over (0, L)…”

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confidence: 99%

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Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

Kong¹,

Wang²

2022

DCDS

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<p style='text-indent:20px;'>One of the most impressive findings in chemotaxis is the aggregation that randomly distributed bacteria, when starved, release a diffusive chemical to attract and group with others to form one or several stable aggregates in a long time. This paper considers pattern formation within the minimal Keller–Segel chemotaxis model with a focus on the stability and dynamics of its multi-spike steady states. We first show that any steady-state must be a periodic replication of the spatially monotone one and they present multi-spikes when the chemotaxis rate is large; moreover, we prove that all the multi-spikes are unstable through their refined asymptotic profiles, and then find a fully-fledged hierarchy of free entropy energy of these aggregates. Our results also complement the literature by finding that when the chemotaxis is strong, the single boundary spike has the least energy hence is the most stable, the steady-state with more spikes has larger free energy, while the constant has the largest free energy and is always unstable. These results provide new insights into the model's intricate global dynamics, and they are illustrated and complemented by numerical studies which also demonstrate the metastability and phase transition behavior in chemotactic movement.</p>

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confidence: 99%

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Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

Kong¹,

Wang²

2022

DCDS

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“…In [2], we estabished the uniqueness of the spike solution, provided its rigorous asymptotic expansion, and proved that the solution is locally exponentially stable. The associated eigenvalue problem is reinvestigated in [16] in a general setting with a systematic method.…”

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confidence: 99%

Spectral analysis for stability of bubble steady states of a Keller–Segel's minimal chemotaxis model

Zhang

Chen

Hao

et al. 2017

Journal of Mathematical Analysis and Applications

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A number of techniques, some of which are novel, are introduced to develop a systematic method to study a set of eigenvalue problems arising from the stability analysis of bubble steady states of a Keller-Segel's minimal chemotaxis model. Estimates of the eigenvalue with largest real part of an elliptic system without variational structure and the second eigenvalue of a corresponding subproblem possessing variational structure are obtained. These estimates provide critical information about the stability of the bubble steady state with respect to the time relaxation parameter; in particular, it is shown that the stability decreases to zero as the relaxation parameter goes to infinity.

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“…Using a phase plane analysis, Chen, Hao, Wang, Wu, and Zhang established in [8] the existence, uniqueness, and local exponential stability of the spike steady state solution, together with rigorous arbitrary high order internal and boundary layer asymptotic expansions. To obtain more detailed information on the stability of the spike solution, we reinvestigated in [38] the associated eigenvalue problem in a general setting with a systematic method.…”

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Dynamics of spike in a Keller-Segel's minimal chemotaxis model

Chen¹,

Hao²,

Lai³

et al. 2017

Discrete &Amp; Continuous Dynamical Systems - A

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The dynamics are studied for the Keller-Segel's minimal chemotaxis model τ ut = (ux − kuvx)x, vt = vxx − v + u on a bounded interval with homogeneous Neumann boundary conditions, where τ 0 and k 1 are parameters and the total mass of u is scaled to be one. In general, the dynamics can be divided into three stages: the first stage is very short in which u quickly becomes a delta like function with mass concentrated near the point of global maximum of v; in the second stage, the point of the global maximum of v drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the profile of the solution evolves to a steady state profile. This paper considers a special case in which the relaxation parameter τ is set to be zero, so the first stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.

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An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model

Cited by 5 publications

References 13 publications

Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model

Spectral analysis for stability of bubble steady states of a Keller–Segel's minimal chemotaxis model

Dynamics of spike in a Keller-Segel's minimal chemotaxis model

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