Singular perturbation and stationary solutions of parabolic equations in Gauss-Sobolev spaces

The paper is concerned with a class of parabolic equations in a Gauss-Sobolev space containing a small parameter " > 0. They degenerate into elliptic equations as " tends to zero. It is proven that, under appropriate conditions, the solution to the Cauchy problem for such a parabolic equation converges, as " # 0, to a limit that satisfies the reduced elliptic equation. This singular perturbation problem is shown to be closely related to the stationary solution of the parabolic problem as t ! 1. An application of this result to the asymptotic evaluation of a certain functional integral is given.