Justification of the exact asymptotics of the fundamental solution for a degenerate parabolic equation with a small parameter

Abstract

AbstractIn this paper, we propose a scheme for constructing the asymptotics of the fundamental solution of a linear degenerate parabolic equation with a small parameter. The asymptotics is constructed using the operator representation of the Dirac delta function and the non‐oscillating Wentzel‐Kramers‐Brillouin (WKB) method. In addition, a method to justify the obtained asymptotics by proving the convergence of the formal series that arises when using the WKB method is shown. Several examples of degenerate parabolic equations are considered, for which a fundamental solution is constructed, and its main properties are indicated.