Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations

Ducrot, Arnaud; Giletti, Thomas; Matano, Hiroshi

doi:10.1090/s0002-9947-2014-06105-9

Cited by 102 publications

(159 citation statements)

References 20 publications

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“…The attractivity of pulsating travelling waves was not known before, even if, as already mentioned, it has been announced by Ducrot, Giletti and Matano in a parallel work [9]. Note that we do not need any hypothesis involving the linearization of equation (E) near the steady state u = 1.…”

Section: Random Stationary Ergodic Heterogeneitiesmentioning

confidence: 88%

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Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

Nadin

2015

Ann. Inst. H. Poincaré Anal. Non Linéaire

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Section: Random Stationary Ergodic Heterogeneitiesmentioning

confidence: 88%

“…Let us just mention that the attractivity of pulsating travelling waves has been proved by Xin [31] in the ignition-type framework but is still an open problem in the monostable one. However, Ducrot, Giletti and Matano [9] have announced some results in this direction.…”

Section: Pulsating Travelling Waves For Periodic Equationsmentioning

confidence: 99%

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

Nadin

2015

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…for any δ > 0 (see [8] for further results of this nature). In particular, they do not have bounded widths.…”

Section: Solutions Of the Cauchy Problem With Bounded Widthsmentioning

confidence: 91%

“…We also let f 1 : [0, 1] → [0, ∞) be any Lipschitz function with constant K such that 8) which is also pure ignition if θ > 0 in (H) and pure monostable (i.e. f (0) = f (1) = 0 and f (u) > 0 for u ∈ (0, 1)) otherwise.…”

mentioning

confidence: 99%

Propagation of Reactions in Inhomogeneous Media

Zlatoš

2016

Comm Pure Appl Math

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Abstract. Consider reaction-diffusion equation u t = ∆u + f (x, u) with x ∈ R d and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0, 1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the super-level sets {u ≥ ε} and {u ≥ 1 − ε} remains uniformly bounded in time for each ε ∈ (0, 1). Thus, u remains uniformly in time close to the characteristic function of {u ≥ 1 2 } in the sense of Hausdorff distance of super-level sets. We also show that {u ≥ 1 2 } expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x-independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 involves construction of a counter-example for which this fails.Such results were before known for d = 1 but are new for general non-periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria u − < u + of the PDE, and to solutions not necessarily satisfying u − ≤ u ≤ u + .

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“…For further stability results and attractivity of the fronts with minimal speeds, we refer to [10,18,23].…”

Section: Pulsating Frontsmentioning

confidence: 99%

Persistence and propagation in periodic reaction-diffusion models

Hamel

Roques

2014

Tamkang J. Math.

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Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations

Abstract: International audienc

Cited by 102 publications

References 20 publications

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

Propagation of Reactions in Inhomogeneous Media

Persistence and propagation in periodic reaction-diffusion models

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