Stability of localized structures in non-local reaction-diffusion equations

Bose, Amitabha; Kriegsmann, Gregory A.

doi:10.4310/maa.1998.v5.n4.a2

Cited by 17 publications

(31 citation statements)

References 11 publications

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“…Hence, this qualitative feature of the spectrum is preserved for finite values of D until D crosses below D 2 . The k-spike hot-spot solution of the scalar nonlocal microwave heating model of [28][29][30] has k − 1 unstable real eigenvalues.…”

Section: Theoretical Results On the Spectrum: Large Eigenvaluesmentioning

confidence: 99%

“…A similar nonlocal model has been analyzed in detail in the context of hot-spot patterns arising in the microwave heating of ceramic materials (cf. [28][29][30]). The first observation and formal analysis of growing alternating-sign fluctuations for an activator-type variable in a reaction-diffusion system was given in [7] and [8] (see Sections 14.4.8,15.3;Figures 14.13 and 14.16 of [7]).…”

Section: Introductionmentioning

confidence: 99%

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The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime

2005

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In a singularly perturbed limit of small diffusivity ε of one of the two chemical species, equilibrium spike solutions to the Gray-Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k-spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k-spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues λ in the spectrum of the linearization. These eigenvalues, which have the property that λ = O(1) as ε → 0, govern the stability of the solution on an O(1) time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric k-spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset and 9600 Garsington Road, Oxford, OX4 2DQ, UK. 22T. Kolokolnikov et al. of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer-Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.

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Section: Theoretical Results On the Spectrum: Large Eigenvaluesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime

2005

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“…The study of bifurcations and solution stability in nonlocal equations is an area of substantial current interest; see, for example, [4,12] and the references therein. In this paper such difficulties are largely bypassed since, although (2.5) is nonlocal, the linear stability analysis of w(x) ≡ 0 remains a local problem.…”

Section: Model Equationsmentioning

confidence: 99%

Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Dawes¹

2008

SIAM J. Appl. Dyn. Syst.

114

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Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical "pattern-forming" instability and intertwine in a manner often referred to as "homoclinic snaking." In this paper we consider an extension of this paradigm where, in addition to the pattern-forming instability (with nonzero wavenumber), a large-scale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal Ginzburg-Landau equation to describe the formation of localized states. Our results show that the existence of the large-scale mode substantially enlarges the region of parameter space where localized states exist and are stable.

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“…The motivation for the extension of the existing boundary feedback design methodologies to the case of nonlinear parabolic PDEs with nonlocal terms is strong. Nonlinear and possibly nonlocal parabolic PDEs arise in many physical problems: see for instance [2,3,15,22,23]. More specifically, a nonlinear and possibly nonlocal PDE may be an equivalent description of a system of parabolicelliptic PDEs.…”

Section: Introductionmentioning

confidence: 99%

Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of One-Dimensional Semilinear Parabolic PDEs

Karafyllis¹,

Krstić²

2019

SIAM J. Control Optim.

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This paper presents a novel methodology for the design of boundary feedback stabilizers for 1-D, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can be applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition. The nonlinearities may contain nonlocal terms. Two different types of boundary feedback stabilizers are constructed: a linear static boundary feedback and a nonlinear dynamic boundary feedback. It is also shown that there are fundamental limitations for feedback design in the parabolic case: arbitrary gain assignment is not possible by means of boundary feedback. An example with a nonlocal nonlinear term illustrates the applicability of the proposed methodology.

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Stability of localized structures in non-local reaction-diffusion equations

Cited by 17 publications

References 11 publications

The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime

The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime

Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of One-Dimensional Semilinear Parabolic PDEs

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