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

Emeric Bouin (1) , Jérôme Coville (2) , Guillaume Legendre (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 BioSP - Biostatistique et Processus Spatiaux

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.

Mots clés: generic nonlocal dispersion operators; fractional Laplace operator; convolution operator; acceleration; level lines

Domaines: Mathématiques [math]

Emeric Bouin, Jérôme Coville, Guillaume Legendre. Sharp exponent of acceleration in general nonlocal equations with a weak Allee effect. 2022. ⟨hal-03452141v2⟩