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

This paper is concerned with the linear ODE in the form y′(t) = λρ(t)y(t) + b(t), λ < 0 which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function ρ(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.