A critical case of stability in a free boundary problem

Bistable reaction-diffusion equations are known to admit one-dimensional travelling waves which are globally stable to one-dimensional perturbations-Fife and McLeod [7]. These planar waves are also stable to two-dimensional perturbations-Xin [30], Levermore-Xin [19], Kapitula [16]-provided that these perturbations decay, in the direction transverse to the wave, in an integrable fashion. In this paper, we first prove that this result breaks down when the integrability condition is removed, and we exhibit a large-time dynamics similar to that of the heat equation. We then apply this result to the study of the large-time behaviour of conical-shaped fronts in the plane, and exhibit cases where the dynamics is given by that of two advection-diffusion equations.

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“…We note that a result similar to [ii] was already proved by Brauner-Hulshof-Lunardi [4], in the case of the following free boundary problem:…”

Section: Resultsmentioning

confidence: 65%

Nontrivial large-time behaviour in bistable reaction–diffusion equations

Roquejoffre

Roussier-Michon

2008

Annali di Matematica

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“…It transpires, however, that the cellular instability may be described by a free interface problem [9][4] [3]. Moreover, near the instability threshold it is possible to (asymptotically) separate the spatial and the temporal coordinates, and further reduce the system to a single geometrically invariant surface dynamics equation.…”

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confidence: 99%

On the κ - θ model of cellular flames: Existence in the large and asymptotics

Brauner¹,

Frankel²,

Hulshof³

et al. 2008

Discrete &Amp; Continuous Dynamical Systems - S

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Abstract. We consider the κ − θ model of flame front dynamics introduced in [6]. We show that a space-periodic problem for the latter system of two equations is globally well-posed. We prove that near the instability threshold the front is arbitrarily close to the solution of the Kuramoto-Sivashinsky equation on a fixed time interval if the evolution starts from close configurations. The dynamics generated by the model is illustrated by direct numerical simulation.2000 Mathematics Subject Classification. Primary: 35K55, 35B25; Secondary: 80A25.

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“…The decomposition (2.7) may look a bit strange; it is similar to the decompositions used in the papers [3,4,5] and in others to study stability problems. It is crucial in our analysis because it lets us decouple system (2.8), expressing s in terms of w. Indeed, the boundary condition u − g 0 = 0 on ∂Ω t is rewritten as…”

Section: The Change Of Coordinatesmentioning

confidence: 94%

“…In the unbounded domain case, C 2+α initial data near a planar travelling wave solution of (1.4) have been considered in [5]. The wave turns out to be orbitally stable, but the discussion is not trivial, because in dimension N ≥ 2 this is a very critical case of stability.…”

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confidence: 99%

“…Indeed, the uniqueness results available up to now concern only particular situations, such as radially symmetric solutions of (1.4), studied in [10], and solutions in cylinders, for initial data which are monotonic in the direction of the axis of the cylinder (see [15] in dimension N = 2 and [12] in any dimension), or in the direction orthogonal to the axis of the cylinder (see [1], for the two-phase case). Moreover, the above mentioned papers [3,5] also give uniqueness results in parabolic Hölder spaces, but only for solutions close to the special solutions considered.…”

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confidence: 99%

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