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March  2008, 1(1): 127-137. doi: 10.3934/krm.2008.1.127

Fast-reaction limit for the inhomogeneous Aizenman-Bak model

José A. Carrillo 1, , Laurent Desvillettes 2, and  Klemens Fellner 3,

1. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
2. 

ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex
3. 

Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090 Wien

Received  October 2007 Revised  October 2007 Published  February 2008

Solutions of the spatially inhomogeneous diffusive\linebreak Aizenmann-Bak model for clustering within a bounded domain with homogeneous Neumann boundary conditions are shown to stabilize, in the fast reaction limit, towards local equilibria determined by their monomer density. Moreover, the sequence of monomer densities converges to the solution of a nonlinear diffusion equation whose nonlinearity depends on the size-dependent diffusion coefficient. Initial data are assumed to be integrable, bounded and with a certain number of moments in size. The number density of clusters for the solutions is assumed to verify uniform bounds away from zero and infinity independently of the scale parameter.
Keywords: nonlinear-diffusion equations, entropy-based estimates, fast-reaction limit, duality arguments., coagulation-fragmentation equation.
Mathematics Subject Classification: 35B40, 35Q72, 82D6.
Citation: José A. Carrillo, Laurent Desvillettes, Klemens Fellner. Fast-reaction limit for the inhomogeneous Aizenman-Bak model. Kinetic and Related Models, 2008, 1 (1) : 127-137. doi: 10.3934/krm.2008.1.127
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