一类非线性抛物型对流扩散方程的相似解

EXPLICIT AND SIMILARITY SOLUTIONS FOR CERTAINNONLINEAR PARABOLIC DIFFUSION-CONVECTIONEQUATIONS

本文讨论对流扩散方程:的相似源型解．由于方程具有对流项且初始值为Ｄｉｒａｃ测度,特别当ｍ＞１时方程在{ｕ＝０}处退化,给求解造成实质的困难．本文基于作者以往研究的有关结果,应用特殊函数理论和微分方程极值原理等技巧,讨论了形如ｕ(ｘ,ｔ)＝ｔ－αｆ(ｘｔ－β)的相似解存在性、唯一性及其数学结构特征．指出当方程存在这种特殊解时ｍ与ｎ应满足某种关系,从量纲分析上看,此种条件是必要的．

Abstract In this paper we finded several explicit and similarity solution to the so-callednonlinear Fokker-Planck equationsut = (um)xx + (um)xfor (x, t) ∈ S with initial datum u(x, 0) = δ(x), where the physical constants m ≥ 1 and n ≥ 0and δ(x) is Dirac measure. Such kind of source-type solutions is an important class of selfsimilarsolutions since they provide a preview of much of the theory of Fokker-Planck equations. Thiswork will be foundation for further researching on such fields.