Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

TL;DR

This paper investigates elliptic operators with unbounded coefficients in unbounded domains, establishing existence, uniqueness, and properties of solutions to associated parabolic equations, and analyzing their evolution families and measures.

Contribution

It introduces a framework for solving parabolic equations with unbounded coefficients, including gradient estimates and the existence of evolution systems of measures.

Findings

Unique bounded classical solutions for the Cauchy problem

Gradient estimates for the evolution family

Existence of evolution systems of measures

Abstract

We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times\CR^d$ for some unbounded interval $I\subset\CR$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot)=f\in C_b(\CR^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family $\{G(t,s)\}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $\{G(t,s)\}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.