THE ASYMPTOTIC CHARACTERISTICS OF THE SOLUTIONS OF THE DIFFUSION EQUATION WITH A NONLINEAR SINK - A RENORMALIZATION-GROUP APPROACH

A non-linear generalization of the diffusion equation, which describes the mass or heat transfer accompanied with chemical reactions, is use d to consider the spreading of an initially localized distribution. Th e use of a renormalization group method enabled the nature of the solu tion to be analysed for long times and two characteristics of its asym ptotic behaviour to be distinguished. When the dimension of the space is greater than a certain critical value, a state of asymptotic freedo m is attained for which the role of non-linearity is small and the evo lution of the density distribution is governed by diffusion processes. When the dimension is less than the critical value, the non-linear te rm remains substantial for long periods of time and a state of incompl ete self-similarity of the evolution of the density distribution is es tablished. The exponent of the exponential dependence of the radius of the diffusion spot on time is calculated for this case. The relation between the renormalization group method and perturbation theory and d ifficulties in substantiating the method when applied to a given probl em are discussed. (C) 1998 Elsevier Science Ltd. All rights reserved.