We study the large time behaviour of the Fisher-KPP equation ∂tu = ∆u+u−u2 in spatial dimension N, when the initial datum is compactly supported. We prove the existence of a Lipschitz function s∞ of the unit sphere, such that u(t, x) approaches, as t goes to infinity, the function Uc∗ ( |x| − c∗t + Nc+∗2 lnt + s∞(|xx| )) , where Uc∗ is the 1D travelling front with minimal speed c∗ = 2. This extends an earlier result of Gärtner.
Sharp large time behaviour in n-dimensional Fisher-KPP equations / Roquejoffre, J. -M.; Rossi, L.; Roussier-Michon, V.. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 39:12(2019), pp. 7265-7290. [10.3934/dcds.2019303]
Sharp large time behaviour in n-dimensional Fisher-KPP equations
Rossi L.;
2019
Abstract
We study the large time behaviour of the Fisher-KPP equation ∂tu = ∆u+u−u2 in spatial dimension N, when the initial datum is compactly supported. We prove the existence of a Lipschitz function s∞ of the unit sphere, such that u(t, x) approaches, as t goes to infinity, the function Uc∗ ( |x| − c∗t + Nc+∗2 lnt + s∞(|xx| )) , where Uc∗ is the 1D travelling front with minimal speed c∗ = 2. This extends an earlier result of Gärtner.| File | Dimensione | Formato | |
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