Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil

TL;DR

This paper investigates the long-term behavior of solutions to linear differential equations with periodic coefficients and drift terms, modeling soil carbon storage, and extends the analysis to heat equations with numerical illustrations.

Contribution

It provides new insights into how drifted periodic coefficients influence the asymptotic behavior of solutions in soil carbon models and heat equations.

Findings

Linear drift in coefficients causes linear drift in solutions

Results extend from ODE to heat equations

Numerical examples illustrate theoretical findings

Abstract

This paper is concerned with the linear ODE in the form $y'(t)=\lambda\rho(t)y(t)+b(t)$, $\lambda <0$ which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function $\rho(t)$, a linear drift in the coefficient $b(t)$ involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.