Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

TL;DR

This paper investigates the long-term behavior of certain non-autonomous parabolic equations with unbounded, time-periodic coefficients, extending previous results on Markov semigroups and analyzing spectral properties in $L^p$ spaces.

Contribution

It generalizes and improves existing asymptotic behavior results for Markov semigroups with invariant measures and explores spectral properties of the associated operators.

Findings

Extended asymptotic behavior results for non-autonomous parabolic equations.

Analyzed spectral properties of the parabolic operator in $L^p$ spaces.

Provided new insights into the long-term dynamics of equations with unbounded coefficients.

Abstract

We study asymptotic behavior in a class of non-autonomous second order parabolic equations with time periodic unbounded coefficients in $\mathbb R\times \mathbb R^d$. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator $u\mapsto {\cal A}(t) u - u_t$ in suitable $L^p$ spaces.