# Fundamental solution for Cauchy initial value problem for parabolic PDEs with discontinuous unbounded first-order coefficient at the origin. Extension of the classical parametrix method

## TL;DR

This paper establishes the existence and estimates of a fundamental solution for a class of parabolic PDEs with discontinuous, unbounded coefficients at the origin, extending Levi's classical parametrix method.

## Contribution

It extends the classical parametrix method to handle parabolic PDEs with discontinuous, unbounded coefficients at the origin, proving fundamental solution existence and estimates.

## Key findings

- Existence of fundamental solution for PDEs with discontinuous coefficients
- Non-asymptotic, rapidly decreasing estimates at infinity
- Extension of Levi's classical parametrix method

## Abstract

We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method provided by E.E. Levi.