Singular Hamilton-Jacobi equation modeling the tail problem

Abstract : In this paper we study the long time-long range behavior of reaction diffusion equations with negative square root -type reaction terms. In particular we investigate the exponential behavior of the solutions after a standard hyperbolic scaling. This leads to a Hamilton-Jacobi variational inequality with an obstacle that depends on the solution itself and defines the open set where the limiting solution does not vanish. Counter-examples show a nontrivial lack of uniqueness for the variational inequality depending on the conditions imposed on the boundary of this open set. Both Dirichlet and state constraints boundary conditions play a role. When the competition term does not change sign, we can identify the limit, while, in general, we find lower and upper bounds for the limit. Although models of this type are rather old and extinction phenomena are as important as blow-up, our motivation comes from the so-called ''tail problem'' in population biology. One way to avoid meaningless exponential tails, is to impose extra-mortality below a given survival threshold. Our study shows that the precise form of this extra-mortality term is asymptotically irrelevant and that, in the survival zone, the population profile is impacted by the survival threshold (except in the very particular case when the competition term is non-positive).
Type de document :
Article dans une revue
SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2012, 44 (6), pp.4297-4319
Liste complète des métadonnées

Littérature citée [18 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-00503154
Contributeur : Guy Barles <>
Soumis le : mardi 16 août 2011 - 09:47:56
Dernière modification le : jeudi 21 mars 2019 - 13:03:33
Document(s) archivé(s) le : vendredi 25 novembre 2011 - 11:10:36

Fichiers

MBPS-review-final.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-00503154, version 2
  • ARXIV : 1007.2802

Citation

Sepideh Mirrahimi, Guy Barles, Benoît Perthame, Panagiotis E. Souganidis. Singular Hamilton-Jacobi equation modeling the tail problem. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2012, 44 (6), pp.4297-4319. 〈hal-00503154v2〉

Partager

Métriques

Consultations de la notice

593

Téléchargements de fichiers

280