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 ...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.展开更多
基金Beijing Natural Sciences Foundation (Grant Nos. 1992002 and 1002004) Beijing Education Committee Foundation, and partially supported by the National Youth Foundation of China.
文摘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.
