Asymptotics of Fundamental Solution of Cauchy Problem for Parabolic Equation with Small Parameter and Degeneration

TL;DR

This paper develops asymptotic formulas for the fundamental solution of a degenerate parabolic equation with a small parameter, using advanced methods including WKB and symplectic geometry to analyze Green's function characteristics.

Contribution

It extends previous asymptotic analysis to degenerate equations and provides detailed insights into the Green's function via symplectic geometry techniques.

Findings

Asymptotic formulas for fundamental solutions are derived.

Green's function characteristics are analyzed using symplectic geometry.

The work generalizes previous results to degenerate parabolic equations.

Abstract

In this paper, the method of constructing the asymptotics of the fundamental solution of the Cauchy problem for a degenerate linear parabolic equation with small diffusion is considered. Based on the results obtained in \cite{dn}, the study extends them over the case of a degenerate equation. As in \cite{dn}, the main technique that allows us to switch from pseudo-differential equations to partial differential equations is the non-oscillating WKB method. A distinctive feature of this work is a more detailed consideration on the characteristics of the Green's function in terms of symplectic geometry. The most significant intermediate result is presented as a theorem on the properties of the fundamental solution.