On the asimptotic behavior of solutions of the Cauchy problem for parabolic equations with time periodic coefficients

TL;DR

This paper investigates the long-term behavior of solutions to second-order parabolic equations with periodic coefficients, linking the problem to degenerate diffusion processes on a combined circle and Euclidean space.

Contribution

It introduces a reduction of the asymptotic analysis to degenerate diffusion processes on a product space, providing new insights into the behavior of solutions with time-periodic coefficients.

Findings

Asymptotic behavior characterized for solutions as t approaches infinity.

Reduction to degenerate diffusion processes simplifies analysis.

Provides a framework for understanding periodic coefficient effects.

Abstract

We are considering the asimptotic behavior as $t\to\infty$ of solutions of the Cauchy problem for parabolic second order equations with time periodic coefficients. The problem is reduced to considering degenerate time-homogeneous diffusion processes on the product of a unit circle and Euclidean space.