Existence and qualitative properties of multidimensional conical bistable fronts

Hamel, François; Monneau, Régis; Roquejoffre, Jean‐Michel

doi:10.3934/dcds.2005.13.1069

Cited by 113 publications

(78 citation statements)

References 25 publications

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“…We mention that even the classical Allen-Cahn equation can exhibit fronts with corners in the plane [16,24,28]. These fronts, however, are fundamentally different from the fronts discussed here.…”

Section: Planar Frontsmentioning

confidence: 41%

Stability of Front Solutions of the Bidomain Equation

Mori

Matano

2016

Comm Pure Appl Math

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The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen-Cahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical AllenCahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen-Cahn equation in striking contrast to the classical or anisotropic Allen-Cahn equations. We identify two types of instabilities, one with respect to long-wavelength perturbations, the other with respect to medium-wavelength perturbations. Interestingly, whether the front is stable or unstable under longwavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate-wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate-wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.

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Section: Planar Frontsmentioning

confidence: 41%

Stability of Front Solutions of the Bidomain Equation

Mori

Matano

2016

Comm Pure Appl Math

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“…In the beginning of the proof of Theorem 2.8, we introduced a V -shaped front φ(x 1 , x 2 − ct) solving (1.1) in R 2 with c = c f / sin α. Similarly, since 2α − π/2 ∈ (0, π/2), it follows from [36,51] Observe first that (5.32) implies that u(0, x 1 , x 2 ) ≥ max φ f (−x 1 sin(2α) − x 2 cos(2α)), φ f (x 1 sin(2α) − x 2 cos(2α)) =: u 0 (x 1 , x 2 ) for all (x 1 , x 2 ) ∈ R 2 . Let u be the solution of the Cauchy problem associated to (1.1) in R 2 with initial condition u 0 at time t = 0.…”

Section: (527)mentioning

confidence: 84%

“…as z − ψ(r) → −∞ (resp. + ∞), c = c f sin α and ψ ′ (+∞) = cot α, (1.6) for some C 1 function ψ : [0, +∞) → R, see [36,51] and the joint figure. When c f < 0, the same result holds after changing the roles of the limits 0 and 1.…”

Section: Introductionmentioning

confidence: 99%

“…A large literature has been devoted to the study of these axisymmetric and non-axisymmetric fronts in the recent years. For symmetry, uniqueness, stability and further qualitative properties of these traveling fronts, we refer to [34,36,37,51,52,56,66,68] (see also [39] for the existence of conical-shaped fronts for some systems of reaction-diffusion equations and for angles α close to π/2). When N ≥ 2 and f fulfills (1.4) with c f = 0, fronts u(t, x) = φ(|x ′ |, x N − ct) with conicalshaped level sets cannot exist anymore, see [34].…”

Section: Introductionmentioning

confidence: 99%

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Bistable transition fronts inRN

Hamel

2016

Advances in Mathematics

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This paper is chiefly concerned with qualitative properties of some reaction-diffusion fronts. The recently defined notions of transition fronts generalize the standard notions of traveling fronts. In this paper, we show the existence and the uniqueness of the global mean speed of bistable transition fronts in R N . This speed is proved to be independent of the shape of the level sets of the fronts. The planar fronts are also characterized in the more general class of almost-planar fronts with any number of transition layers. These qualitative properties show the robustness of the notions of transition fronts. But we also prove the existence of new types of transition fronts in R N that are not standard traveling fronts, thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions.

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“…Besides, in R N with N ≥ 2, more general traveling fronts exist, which have non-planar level sets. For instance, conical-shaped axisymmetric non-planar fronts are known to exist for some f , see [7,16,25]. Fronts with non-axisymmetric shapes, such as pyramidal fronts, are also known to exist, see [34,36].…”

Section: Introductionmentioning

confidence: 99%

Propagating speeds of bistable transition fronts in spatially periodic media

Guo

2018

Calc. Var.

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This paper is concerned with the propagating speeds of transition fronts in R N for spatially periodic bistable reaction-diffusion equations. The notion of transition fronts generalizes the standard notions of traveling fronts. Under the a priori assumption that there exist pulsating fronts for every direction e with nonzero speeds, we show some continuity and differentiability properties of the front speeds and profiles with respect to the direction e. Finally, we prove that the propagating speed of any transition front is larger than the infimum of speeds of pulsating fronts and less than the supremum of speeds of pulsating fronts.

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Existence and qualitative properties of multidimensional conical bistable fronts

Cited by 113 publications

References 25 publications

Stability of Front Solutions of the Bidomain Equation

Stability of Front Solutions of the Bidomain Equation

Bistable transition fronts inRN

Propagating speeds of bistable transition fronts in spatially periodic media

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