The asymptotic characteristics of the solutions of the diffusion equation with a non-linear sink: A renormalization group approachстатья
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Дата последнего поиска статьи во внешних источниках: 11 января 2018 г.
Аннотация:A non-linear generalization of the diffusion equation, which describes the mass or heat transfer accompanied with chemical reactions, is used to consider the spreading of an initially localized distribution. The use of a renormalization group method enabled the nature of the solution to be analysed for long times and two characteristics of its asymptotic behaviour to be distinguished. When the dimension of the space is greater than a certain critical value, a state of asymptotic freedom is attained for which the role of non-linearity is small and the evolution of the density distribution is governed by diffusion processes. When the dimension is less than the critical value, the non-linear term remains substantial for long periods of time and a state of incomplete self-similarity of the evolution of the density distribution is established. 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 difficulties in substantiating the method when applied to a given problem are discussed.