Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in $\R^d$

TL;DR

This paper investigates the compactness and long-term behavior of solutions to nonautonomous linear parabolic equations with unbounded, time-periodic coefficients in multi-dimensional space, providing conditions and describing asymptotic measures.

Contribution

It establishes sufficient conditions for the evolution operator to be compact and characterizes the asymptotic behavior via a family of measures solving the Fokker-Planck equation.

Findings

Conditions for compactness of the evolution operator

Description of asymptotic behavior in terms of measures

Connection to solutions of the Fokker-Planck equation

Abstract

We consider a class of second order linear nonautonomous parabolic equations in R^d with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator G(t,s) be compact in C_b(R^d) for t>s, and describe the asymptotic behavior of G(t,s)f as t-s goes to infinity in terms of a family of measures mu_s, s in R, solution of the associated Fokker-Planck equation.