Propagation acceleration in reaction diffusion equations with anomalous diffusions. (3rd March 2021)

• Record Type: Journal Article
• Authors: Coville, Jérôme; Gui, Changfeng; Zhao, Mingfeng
• Is Part Of: Nonlinearity. Volume 34:Number 3(2021)
• Page Start: 1544
• Page End: 1576
• Publication Date: 2021-03-03
• Subjects: traveling wave -- propagation speed -- interface propagation -- reaction diffusion -- fractional Laplacian; 35B51 -- 35K55 -- 35K57 -- 35R09 -- 35R11; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abe17c
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper we consider the propagation speed in a reaction diffusion system with an anomalous Lévy process diffusion, modeled by a nonlocal equation with a fractional Laplacian and a generalized monostable or ignition nonlinearity. Given a typical Heaviside initial datum, we show that the speed of interface propagation displays an algebraic rate behavior in time, in contrast to the known linear rate in the classical model of Brownian motion and the exponential rate in the KPP model with the anomalous diffusion, and depends on the sensitive balance between the anomaly of the diffusion process and the strength of monostable reaction. In particular, for the combustion model with a fractional Laplacian (−Δ) s, we show that the speed of propagation transits continuously from being linear in time, when a traveling wave solution exists for s ∈ (1/2, 1), to being algebraic in time with a power reciprocal to 2 s, when no traveling wave solution exists for s ∈ (0, 1/2).