Asymptotic stability of the critical Fisher–KPP front using pointwise estimates

Faye, Grégory; Holzer, Matt

doi:10.1007/s00033-018-1048-0

Cited by 26 publications

(37 citation statements)

References 18 publications

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“…). Since the translation eigenmode q * in weighted spaces is unbounded at x → +∞, we expect U to decay in time with rate t −3/2 if the following assumption of a marginally stable essential spectrum holds, see [AS21,FH18]:…”

Section: Resultsmentioning

confidence: 99%

“…The actual choice lighten the computation to obtain the GL equation. Conjugation by := ρ * ω kpp is needed so that proposition 4.5 holds: this weight is natural for KPP equation, in the sense that its spatial decay at +∞ copy the one of the translation eigenfunction q * [FH18].…”

Section: Resultsmentioning

confidence: 99%

“…to x ∈ R. At linear level, we expect an algebraic decay: t −3/2 . We follow the approach from [FH18]: we first look at the solution to the linear Cauchy problem (15)…”

Section: Resultsmentioning

confidence: 99%

“…Having kept the extra exponential localization -ω κ1,0 (x)ω κ1,0 (y) when x ≥ 0 or y ≥ 0 -in our estimates of G kpp λ (x, y), the proof of [FH18] straightforwardly adapt with this extra decay. The same goes for the other case x ≤ 0 and y ≤ 0.…”

Section: Construction Of Decaying Ode Solutionsmentioning

confidence: 99%

“…Stability of the critical front q * := q c * is more involved, since the linear essential spectrum can only be marginally stabilized : in the optimal exponentially weighted space, essential spectrum lies at left of the imaginary axis but touches it, due to the presence of absolute spectrum −ξ 2 : ξ ∈ R ⊂ C up to the origin. Last advances in this direction [Gal94,FH18,AS21] state that q * is asymptotically stable with decay rate t −3/2 , and that this rate is optimal. Gallay used renormalization group method; Faye and Holzer relied on point-wise estimates for the resolvent of the full problem; Avery and Scheel reduced the problem to its asymptotic properties through far-field decomposition, and use resolvent estimates on each side.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study

Garénaux¹

2021

Preprint

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We investigate a specific reaction-diffusion system that admits a monostable pulled front propagating at constant critical speed. When a small parameter changes sign, the stable equilibrium behind the front destabilizes, due to essential spectrum crossing the imaginary axis, causing a Turing bifurcation. Despite both equilibrium states are unstable, the front continues to exist, and is shown to be asymptotically stable, against suitably-localized perturbations, with algebraic temporal decay rate t −3/2 . To obtain such decay, we rely on point-wise semigroup estimates, and show that the Turing pattern behind the front remains bounded in time, by use of mode-filters. Contents 1. Introduction 2. Main result 2.1. Notations 2.2. Sketch of the proof 3. Decay of perturbations in fully weighted space 3.1. Essential and point spectrum 3.2. Construction of decaying ODE solutions 3.3. Construction and estimations of the kernel for the resolvent operator 3.4. Linear dynamic 3.5. Nonlinear dynamic 4. Perturbations in partially weighted space are bounded in time 4.1. Mode filters 4.2. Linear dynamic 4.3. Nonlinear dynamic: shadowing the global attractor of the Ginzburg-Landau equation Appendix A. Ginzburg-Landau equation Appendix B. Proof of proposition 3.8 Appendix C. Three equilibrium points Appendix D. Proof of lemma 3.4 Appendix E.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

“…to x ∈ R. At linear level, we expect an algebraic decay: t −3/2 . We follow the approach from [FH18]: we first look at the solution to the linear Cauchy problem (15)…”

Section: Resultsmentioning

confidence: 99%

Section: Construction Of Decaying Ode Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study

Garénaux¹

2021

Preprint

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Spectral stability of the critical front in the extended Fisher-KPP equation

Avery

Garénaux

2023

Z. Angew. Math. Phys.

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Traveling waves of an FKPP-type model for self-organized growth

Florian

2022

J. Math. Biol.

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We consider a reaction–diffusion system of densities of two types of particles, introduced by Hannezo et al. (Cell 171(1):242–255.e27, 2017). It is a simple model for a growth process: active, branching particles form the growing boundary layer of an otherwise static tissue, represented by inactive particles. The active particles diffuse, branch and become irreversibly inactive upon collision with a particle of arbitrary type. In absence of active particles, this system is in a steady state, without any a priori restriction on the amount of remaining inactive particles. Thus, while related to the well-studied FKPP-equation, this system features a game-changing continuum of steady state solutions, where each corresponds to a possible outcome of the growth process. However, simulations indicate that this system self-organizes: traveling fronts with fixed shape arise under a wide range of initial data. In the present work, we describe all positive and bounded traveling wave solutions, and obtain necessary and sufficient conditions for their existence. We find a surprisingly simple symmetry in the pairs of steady states which are joined via heteroclinic wave orbits. Our approach is constructive: we first prove the existence of almost constant solutions and then extend our results via a continuity argument along the continuum of limiting points.

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Asymptotic stability of the critical Fisher–KPP front using pointwise estimates

Cited by 26 publications

References 18 publications

Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study

Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study

Spectral stability of the critical front in the extended Fisher-KPP equation

Traveling waves of an FKPP-type model for self-organized growth

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