Gaussian estimates for fundamental solutions of higher-order parabolic equations with time-independent coefficients

TL;DR

This paper establishes Gaussian upper bounds and regularity estimates for fundamental solutions of higher-order parabolic equations with complex, measurable, and time-independent coefficients, extending classical results to more complex operators.

Contribution

It provides new Gaussian estimates and regularity results for higher-order parabolic equations with complex-valued, time-independent coefficients, broadening the scope of classical theory.

Findings

Gaussian upper bounds for fundamental solutions

Hölder regularity estimates for solutions

Extension of De Giorgi-Moser-Nash theory to higher-order equations

Abstract

We study the De Giorgi-Moser-Nash estimates of higher-order parabolic equations in divergence form with complex-valued, measurable, bounded, uniformly elliptic (in the sense of G$\mathring{a}$rding inequality) and time-independent coefficients. We also obtain Gaussian upper bounds and H\"{o}lder regularity estimates for the fundamental solutions of this class of parabolic equations.