Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations

Roquejoffre, Jean‐Michel; Rossi, Luca; Roussier-Michon, Violaine

doi:10.3934/dcds.2019303

Cited by 19 publications

(31 citation statements)

References 24 publications

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“…To conclude this section, we note that a similar analysis can also be applied to a ball in R n . As in the one-dimensional case, this appears to match the coefficient of the logarithmic correction term in the nonlinear FKPP problem on R n , with compactly supported initial conditions (see [18,24]). .…”

Section: Critical Case In Higher Dimensionsmentioning

confidence: 67%

“…Bramson's logarithmic term (or similar) has been seen to arise in several other circumstances. We note, in particular, the paper [18], by Gärtner, which generalized the result to the multi-dimensional case (see also [24]), and the paper [7], by Berestycki (J. ), Brunet and Derrida, which derived the term in the setting of a linear equation on a semi-infinite interval with a free boundary.…”

Section: Introductionmentioning

confidence: 91%

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Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

Allwright¹

2021

Proceedings of the Edinburgh Mathematical Society

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A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

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Section: Critical Case In Higher Dimensionsmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 91%

Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

Allwright¹

2021

Proceedings of the Edinburgh Mathematical Society

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“…[2,9,31,36,37,55] for extinction/invasion results in terms of the size and/or the amplitude of the initial condition u 0 for various reaction terms f , and to [9,11,34,35,42] for general local convergence and quasiconvergence results at large time. For the invading solutions u (that is, those converging to 1 locally uniformly in R N as t → +∞) with localized initial conditions, further estimates on the location and shape at large time of the level sets have been established in [13,18,27,45,49,51,53]. Lastly, equations of the type (1.1) set in unbounded domains Ω instead of R N and notions of spreading speeds and persistence/invasion in such domains have been investigated in [5,50].…”

Section: Two Main Questionsmentioning

confidence: 99%

“…for some Lipschitz continuous function a defined in S N −1 , see [13,18,45]. Notice that N + 2 = 3 + (N − 1) corresponds to an additional lag by ((N − 1)/c * ) log t, compared with the 1-dimensional case, which is due to the curvature of the level sets inherited from the fact that the initial condition is compactly supported.…”

Section: The Logarithmic Lag In the Kpp Casementioning

confidence: 99%

“…Let u be the solution to (1.1) emerging from a continuous, compactly supported, radially symmetric and non-trivial initial datum u 0 such that 0 ≤ u 0 ≤ u 0 ≤ 1 in R N . We know from [18] or [45,Theorem 1.1] that there exists σ ∈ R such that, for any K > 0, there exists T K > 0 for which there holds In order to show the reverse inequality, we will construct a supersolution v larger than u at time 0, for which we are able to explicitly compute the lag. First of all, owing to (1.50), we can take β ∈ R satisfying lim sup…”

Section: Directional Asymptotic One-dimensional Symmetrymentioning

confidence: 99%

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Spreading speeds and spreading sets of~reaction-diffusion equations

Hamel¹,

Rossi²

2021

Preprint

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This paper is devoted to the study of the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in R N . We first prove a general Freidlin-Gärtner type formula for the spreading speeds of the solutions in any direction. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of independent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of traveling fronts with positive speed. Furthermore, we show some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we prove some asymptotic one-dimensional symmetry properties for the elements of the Ω-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we show some logarithmicin-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The paper also contains some related conjectures and open problems. * This work has been carried out in the framework of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The research leading to these results has also received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 -ReaDi -Reaction-Diffusion Equations, Propagation and Modelling, from the ANR projects NONLOCAL (ANR-14-CE25-0013) and RESISTE (ANR-18-CE45-0019).

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Spreading in Several Space Dimensions

Roquejoffre

2024

Lecture Notes on Mathematical Modelling in the Life Sciences

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Sharp large time behaviour in $ N $-dimensional Fisher-KPP equations

Cited by 19 publications

References 24 publications

Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

Spreading speeds and spreading sets of~reaction-diffusion equations

Spreading in Several Space Dimensions

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