Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

  • We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary conditions by the solution of a suitable stochastic/deterministic differential equation for the average concentration that involves reactions only. An interesting effect occurs, if the noise on the boundary does not change the averaging concentration, but is sufficiently large. Then surprising additional effective reaction terms appear. We focus on systems with polynomial nonlinearities only and give applications to the two dimensional nonlinear heat equation and the cubic auto-catalytic reaction between two chemicals.

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Author:Wael W. MohammedGND, Dirk BlömkerORCiDGND
Frontdoor URL
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2014-05)
Year of first Publication:2014
Publishing Institution:Universität Augsburg
Release Date:2014/08/12
Tag:multi-scale analysis; SPDEs; stochastic boundary conditions; reaction-diffusion equations; fast diffusion limit
GND-Keyword:Reaktions-Diffusionsgleichung; Partielle Differentialgleichung; Randbedingung <Mathematik>; Mehrskalenanalyse
Erschienen in SIAM Journal on Mathematical Analysis, 48, 5, S. 3547-3578,
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand