Global boundedness of the fundamental solution of parabolic equations with unbounded coefficients

TL;DR

This paper establishes an upper bound for the fundamental solution of parabolic equations with unbounded coefficients, extending understanding of their behavior under certain boundedness conditions on the potential.

Contribution

It provides a novel upper bound estimate for the fundamental solution of parabolic PDEs with unbounded coefficients under specific boundedness assumptions.

Findings

Derived an explicit upper bound for the fundamental solution

Extended existing theory to operators with unbounded coefficients

Applicable to a class of parabolic equations with bounded below potentials

Abstract

The purpose of this paper is to obtain an upper bound for the fundamental solution for parabolic Cauchy problem u'=Au, where A is a second order elliptic partial differential operator with unbounded coefficients such that its potential and the potential of the formal adjoint of the operator A are bounded from below.