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Sharp exponent of acceleration in general nonlocal equations with a weak Allee effect

Bouin, Emeric; Coville, Jérôme; Legendre, Guillaume (2021), Sharp exponent of acceleration in general nonlocal equations with a weak Allee effect. https://basepub.dauphine.psl.eu/handle/123456789/22724

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Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-03452141
Date
2021
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
45
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Author(s)
Bouin, Emeric
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Coville, Jérôme cc
Biostatistique et Processus Spatiaux [BioSP]
Legendre, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We study a acceleration phenomenon arising in monostable integro-differential equations with a weak Allee effect. Previous works have shown its occurrence and have given correct upper bounds on the rate of expansion, but precise lower bounds were still missing. In this paper, we provide a sharp lower bound of acceleration for a large class of dispersion operators. Our results cover fractional Laplace operators and standard convolutions in a unified way. To achieve this, we construct a refined sub-solution that captures the expected dynamics of the accelerating solution. We also take advantage of a general flattening estimate for the solution, that we prove along the way and is of independent interest.
Subjects / Keywords
level lines; acceleration; convolution operator; fractional laplacian; generic nonlocal dispersion operators