, Volume 44, Issue 8, pp 984-993

On a nonhomogeneous Burgers’ equation

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Abstract

In this paper the Cauchy problem for the following nonhomogeneous Burgers’ equation is considered : (1)u t +uu x =μu xx kx,xR,t > 0, where µ 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)ϕ t ϕ xx = −x 2 ϕ. 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.