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

Nadin, Grégoire

doi:10.1016/j.anihpc.2014.03.007

Cited by 52 publications

(62 citation statements)

References 31 publications

(88 reference statements)

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“…In a similar way to the approach in [20,17] or [2] we can interpret positive decreasing solutions of (4) as solution of a first order equation. In fact, by Proposition 2.1, if u : (ζ 0 , +∞) → (0, 1), ζ 0 ≥ −∞ is a solution of (4) and satisfies (5), u is a diffeomorphism from (ζ , +∞) to an open interval (0, β) for some ζ ≥ ζ 0 .…”

Section: Existence Of Fast Solutionmentioning

confidence: 92%

“…These concepts involve wave profiles with different speeds and only deal with global solutions. The authors think that when c 2 M > 4f (0), the profile with this speed must be the first profile of a terrace, see [7], and therefore it is a critical solution in the sense of [21,19] arriving to 0.…”

Section: Definition 12mentioning

confidence: 97%

“…As we will point out throughout the paper, some of our proofs are based on ideas and techniques already used by others authors, see [4][5][6]12,15,17,20]. We have preferred to include here all the proofs because in many cases the terminology is different and the hypotheses required in the most of them are hardly comparable.…”

Section: Definition 12mentioning

confidence: 98%

“…Fast solutions are also related to the concepts of critical traveling waves of [21,19] or steeper of [7]. These concepts involve wave profiles with different speeds and only deal with global solutions.…”

Section: Definition 12mentioning

confidence: 99%

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Fast solutions and asymptotic behavior in a reaction–diffusion equation

Arias

Campos

2015

Journal of Differential Equations

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In this paper we deal with the existence of traveling waves solutions (t.w.s.) for the reaction-diffusion equationin a very general setting of reaction terms f with two distinguished stationary states, say 0 and 1. We link the existence of some type of solutions of the second order ODEwith the existence of fast t.w.s. By defining fast solutions for this ODE, we find a value c M > 0 related to the existence of global fast solutions and determine c M through a variational formula. Our results allow us particularly to show that any solution u(x, t) of the reaction-diffusion equation with compactly supported initial data and 0 ≤ u(x, 0) ≤ 1, x ∈ R, satisfies lim t→+∞ u(x + ct, t) = 0, uniformly on compact sets, for all c, |c| > c M .Finally, we connect c M with the minimum speed of propagation of all t.w.s. c * .

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Section: Existence Of Fast Solutionmentioning

confidence: 92%

Section: Definition 12mentioning

confidence: 97%

Section: Definition 12mentioning

confidence: 98%

Section: Definition 12mentioning

confidence: 99%

See 2 more Smart Citations

Fast solutions and asymptotic behavior in a reaction–diffusion equation

Arias

Campos

2015

Journal of Differential Equations

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“…Existence of transition fronts in one-dimensional disordered media (but no long term asymptotics of general solutions) was also proved for bistable reactions which are small perturbations of homogeneous ones by Vakulenko, Volpert [25], for KPP reactions which are (spatially) decaying perturbations of homogeneous ones by Nolen, Roquejoffre, Ryzhik, Zlatoš [19], for general KPP reactions by Zlatoš [30], and for general monostable reactions which are close to KPP reactions by Tao, Zhu, Zlatoš [24]. We also mention results proving existence of a critical front, once some transition front exists, by Shen [22] and Nadin [18].…”

Section: Introductionmentioning

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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Monotonicity of Bistable Transition Fronts in ℝN

Guo

Hamel

2016

J Elliptic Parabol Equ

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

Cited by 52 publications

References 31 publications

Fast solutions and asymptotic behavior in a reaction–diffusion equation

Fast solutions and asymptotic behavior in a reaction–diffusion equation

Propagation of Reactions in Inhomogeneous Media

Monotonicity of Bistable Transition Fronts in ℝN

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