中国科学A辑(英文版)
中國科學A輯(英文版)
중국과학A집(영문판)
SCIENCE IN CHINA
2001年
8期
984-993
,共10页
nonhomogeneous Burgers' equation%explicit solution%large time behavior%Hermite polynomials
In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x ∈ R, t ＞ O, whereμ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) ψt - ψxx =- x2. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly.Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf' s paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.
