The minimal wave speed of a general reaction-diffusion equation with nonlinear advection
type: Item
instance of: scholarly article
title: The minimal wave speed of a general reaction-diffusion equation with nonlinear advection
zbMATH Open document ID: 1475.35102
author: Manjun Ma
author: Chunhua Ou
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2021-08-10T00:00:00Z
review text: In recent years, many researchers paid attention to the linear and nonlinear determinacy of the minimal wave speed of various monostable reaction-diffusion systems. Since all these systems are monotone systems, the comparison principle together with the upper-lower solutions was used as a main tool. We can refer to [the authors, SIAM J. Math. Anal. 51, No. 1, 321--345 (2019; Zbl 1407.35114)]. For the Burgers-KPP-Fisher equation, the minimal speed has already been studied by \textit{J. D. Murray} [Lectures on nonlinear differential-equation models in biology. Oxford: Clarendon Press (1977)], and numerically demonstrated later by \textit{V. Mendez} and \textit{J. Fort} [`Speed of reaction-transport processes'', Phys. Rev. E (3) 64, No. 1, Article ID 011105, 4 p. (2001; \url{doi:10.1103/PhysRevE.64.011105})], but rigorous proof is not obtained. In this paper, the authors considered a general reaction-diffusion equation with a nonlinear advection term, which is an extension of Burgers-KPP-Fisher equation. Here the convection term models the deterministic motion of substances in the direction of its gradient, while the random motion is modeled by the diffusive term. This advection reaction-diffusion equation has neither monotonicity nor variational structure. Thus, the variation method and the comparison principle cannot be applied. Hence, the study on the linear or nonlinear determinacy of the minimal speed is challenging. The authors developed the classical geometrical argument by subtly constructing invariant regions to establish the existence of monotone traveling waves, and derived a necessary and sufficient condition for the nonlinear selection mechanism via a geometrical method coupled with a perturbation argument in an abstract weighted functional space. As applications, they investigated the Burgers-Fisher model, the model with cubic reaction term and the Burger's convection, and subcritical quintic Ginzburg-Landau equation with nonlinear convection, and success…
reviewed by: Guo-Bao Zhang
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 37N25
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 7380413
zbMATH Keywords: linear and nonlinear selection
zbMATH Keywords: minimal wave speed
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: nonlinear advection
zbMATH Keywords: Burgers-KPP-Fisher equation
MaRDI profile type: Publication
full work available at URL: s00033-021-01588-6
OpenAlex ID: W3190112002
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DOI: 10.1007/S00033-021-01588-6
description: scientific article; zbMATH DE number 7380413