On the mean speed of bistable transition fronts in unbounded domains

Guo, Hongjun; Hamel, François; Sheng, Wei-Jie

doi:10.1016/j.matpur.2020.02.002

Cited by 27 publications

(25 citation statements)

References 58 publications

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“…Moreover, in each output branch far from the junction we prove the convergence to the front profile as t → ∞. The latter result, which is not shown in [20], is largely owing to the recent work by [13] in a domain of multiple branches. We develop their argument to the present case.…”

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confidence: 61%

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Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Jimbo¹,

Morita²

2021

DCDS

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We deal with the bistable reaction-diffusion equation in an infinite star graph, which consists of several half-lines with a common end point. The aim of our study is to show the existence of front-like entire solutions together with the asymptotic behaviors as t → ±∞ and classify the entire solutions according to their behaviors, where an entire solution is meant by a classical solution defined for all t ∈ (−∞, ∞). To this end, we give a condition under that the front propagation is blocked by the emergence of standing stationary solutions. The existence of an entire solution which propagates beyond the blocking is also shown.

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confidence: 61%

“…The proof of Theorem 5.1 largely owes to [13], where they discuss the asymptotic behavior of front-like solutions in a domain with multi-cylindrical branches. Here we interpret their argument in our context and modify it suitably for the present case.…”

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confidence: 99%

Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Jimbo¹,

Morita²

2021

DCDS

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“…We start with the following auxiliary lemma that gives the existence of solutions to elliptic equations in large intervals. The proof is based on variational methods, see for instance [10, Theorem A] and [26,Problem (2.25)]. We omit it here.…”

Section: Blocking In the Bistable Patch 2: Proof Of Theorem 211mentioning

confidence: 99%

Propagation and blocking in a two-patch reaction-diffusion model

Hamel¹,

Lutscher²,

Zhang³

2021

Preprint

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This paper is concerned with propagation phenomena for the solutions of the Cauchy problem associated with a two-patch one-dimensional reaction-diffusion model. It is assumed that each patch has a relatively well-defined structure which is considered as homogeneous. A coupling interface condition between the two patches is involved. We first study the spreading properties of solutions in the case when the per capita growth rate in each patch is maximal at low densities, a configuration which we call the KPP-KPP case, and which turns out to have some analogies with the homogeneous KPP equation in the whole line. Then, in the KPP-bistable case, we provide various conditions under which the solutions show different dynamics in the bistable patch, that is, blocking, virtual blocking (propagation with speed zero), or spreading with positive speed. Moreover, when propagation occurs with positive speed, a global stability result is proved. Finally, the analysis in the KPP-bistable frame is extended to the bistable-bistable case.

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“…Since the domains satisfying (1.7)-(1.8) lead to a variety of interesting and non-trivial phenomena, we restrict ourselves to (1.7)-(1.8) throughout the paper. If the domain is a straight cylinder in the direction x 1 (this happens in the case α = 0), then the planar front φ(x 1 − ct) given by (1.4) solves (1.1) (furthermore, up to translation, any transition front connecting 0 and 1 in the sense of Definition 1.1 below is equal to that front, see [26,28]). Here a domain Ω = Ω R,α satisfying (1.7)-(1.8) is straight in its left part only, and the standard planar front φ(x 1 − ct) does not fulfill the Neumann boundary conditions when α > 0.…”

Section: Notationsmentioning

confidence: 99%

“…In [36], a reaction-diffusion model was considered to analyse the effects on population persistence of simultaneous changes in the position and shape of a climate envelope. Recently, the existence and characterization of the global mean speed of transition fronts in domains with multiple cylindrical branches were investigated in [26]. It was proved that the front-like solutions emanating from planar fronts in some branches and propagating completely are transition fronts moving with the planar speed c and eventually converging to planar fronts in the other branches.…”

Section: Introductionmentioning

confidence: 99%

Reaction-diffusion fronts in funnel-shaped domains

Hamel,

Zhang

2021

Preprint

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We consider bistable reaction-diffusion equations in funnel-shaped domains of R N made up of straight parts and conical parts with positive opening angles. We study the large time dynamics of entire solutions emanating from a planar front in the straight part of such a domain and moving into the conical part. We show a dichotomy between blocking and spreading, by proving especially some new Liouville type results on stable solutions of semilinear elliptic equations in the whole space R N . We also show that any spreading solution is a transition front having a global mean speed, which is the unique speed of planar fronts, and that it converges at large time in the conical part of the domain to a well-formed front whose position is approximated by expanding spheres. Moreover, we provide sufficient conditions on the size R of the straight part of the domain and on the opening angle α of the conical part, under which the solution emanating from a planar front is blocked or spreads completely in the conical part. We finally show the openness of the set of parameters (R, α) for which the propagation is complete.

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On the mean speed of bistable transition fronts in unbounded domains

Cited by 27 publications

References 58 publications

Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

Propagation and blocking in a two-patch reaction-diffusion model

Reaction-diffusion fronts in funnel-shaped domains

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