On solvability of the Cauchy problem for second order parabolic equation degenerating in Schrodinger types

TL;DR

This paper studies the solvability of a second order parabolic equation that degenerates into a Schrödinger type in certain regions, providing conditions for existence and explicit solution representations.

Contribution

It establishes existence conditions and constructs explicit integral solutions for a degenerate parabolic-Schrodinger type equation.

Findings

Existence of solutions under specific data conditions

Explicit integral representation of solutions

Analysis of degeneracy from parabolic to Schrödinger type

Abstract

The Cauchy problem is investigated for the parabolic type in the some finite part $[t_0, t_1] \subset [0, \infty)$ of the semi axis $t \in [0, \infty)$ and degenarated to Schrodinger type in the remain part of the same semi axes the second order parabolic equation. The existence of the solution is proved under some conditions on the data and the explicit integral representation is constructed.