The influence of advection on the propagation of fronts in reaction-diffusion equations.(English)Zbl 1073.35113

Berestycki, Henri (ed.) et al., Nonlinear PDEs in condensed matter and reactive flows. Proceedings of the NATO Advanced Study Institute on PDE’s in models of superfluidity, superconductivity and reactive flows, Cargèse, France, 21 June – 3 July 1999. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0972-0/hbk). NATO ASI Ser., Ser. C, Math. Phys. Sci. 569, 11-48 (2002).
Situations are discussed which are represented by a scalar function $$u=u(t,x)$$, and the general homogeneous reaction-diffusion equation is $u_t-\Delta u=f(u)\quad\text{in}\;\mathbb R^N,$ where $$\Delta u$$ is diffusion and $$f(u)$$ the reaction. This problem is well studied by many authors which are mentioned in the paper. If there is an underlying flow $${q}(x)$$ which gives rise to a transport of the scalar $$u$$, one is led to the reaction-diffusion-advection equation: $u_t-\Delta u+q(x)\nabla u=f(u).\tag{1}$
In this paper, the author focuses on recent developments dealing with the advection reaction-diffusion equation (1). One approach to heterogeneous media in the paper is to consider the framework of propagation of periodic media. For random media is referred to the book of Freidlin. In addition, the definition of travelling fronts is given and some new results regarding the speed of travelling fronts in shear flows with large advection are presented. Moreover, the pulsating travelling fronts is defined and is called periodic travelling fronts. Finally, several open problems are given.
For the entire collection see [Zbl 1028.00035].

MSC:

 35K57 Reaction-diffusion equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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