On the behavior of a nonstationary Poiseuille solution as $t\to\infty$

A nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution to an inverse problem for the heat equation. The behavior as $t\to\infty$ of the nonstationary Poiseuille solution corresponding to the prescribed flux $F(t)$ of the velocity field is studied. In particular, it is proved that if the flux $F(t)$ tends exponentially to a constant flux $F_*$ then the nonstationary Poiseuille solution tends exponentially as $t\to\infty$ to the stationary Poiseuille solution having the flux $F_*$.