Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay

Matthieu Alfaro (1) , Arnaud Ducrot (2, 3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 ANUBIS - Tools of automatic control for scientific computing, Models and Methods in Biomathematics
3 IMB - Institut de Mathématiques de Bordeaux

We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial _t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with {\it slow exponential decay}. We prove that the sharp interface limit moves by a constant speed, which dramatically depends on the tails of the initial data. By performing a fine analysis of both the generation and motion of interface, we provide a new estimate of the thickness of the transition layers.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay. Discrete and Continuous Dynamical Systems - Series B, 2011, pp.16 (2011), 15-29. ⟨10.3934/dcdsb.2011.16.15⟩. ⟨hal-00472212⟩