Extremal case of parabolic differential equations having discontinuous unbounded coefficients. Existence of fundamental solution for an initial Cauchy problem. Parametrix method

TL;DR

This paper establishes the existence of a fundamental solution for a class of parabolic PDEs with discontinuous, unbounded coefficients using an extended parametrix method, providing explicit estimates for the solution.

Contribution

It extends the classical parametrix method to handle parabolic equations with non-integrable, unbounded discontinuous coefficients at the origin.

Findings

Existence of fundamental solution proven for the specified PDE class.

Provided explicit non-asymptotic estimates for the fundamental solution.

Extended the classical parametrix method to more singular coefficient cases.

Abstract

We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded discontinuity for coefficient before a first order derivative. We give also the non-asymptotic rapidly decreasing at infinity estimate for these function. We extend the classical parametrix method offered by E.E.Levi.