On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder

Abstract
The work focuses on the behaviour at infinity of solutions to second order elliptic equation with first order terms in a semi-infinite cylinder. Neu- mann's boundary condition is imposed on the lateral boundary of the cylinder and Dirichlet condition on its base. Under the assumption that the coefficients stabilize to a periodic regime, we prove the existence of a bounded solution, its stabilization to a constant, and provide necessary and sufficient condition for the uniqueness. 1. Introduction. This work deals with the behaviour at infinity of solutions to stationary convection-diffusion equations defined in a semi-infinite cylinder. We assume that Neumann boundary condition is imposed on the lateral boundary of the cylinder, and that the coefficients of the equation are periodic along the cylinder axis or stabilize at the exponential rate to a periodic regime for asymptotically large axial distance. Under these assumptions we study the existence and uniqueness of a bounded solution, and its stabilization to a constant at infinity. The question of validity of the Saint-Venant and Phragmen-Lindelof principles, as well as other questions related to the behaviour at infinity of solutions to elliptic equations and systems of equations, received a lot of attention of mechanicians and mathematicians starting from the beginning of 20th century. A number of rigorous mathematical works are devoted to this subject. Dirichlet and Neumann boundary value problems in a cylindrical domains for second order linear elliptic equations in divergence form were studied by many authors. Early contributions include (10), (6) and (7) which contain results like Saint-Venant's prin- ciple for special classes of Neumann problems. As to the later works on this topic, we mention just some of them closely related to the present paper. In (14) an equation in divergence form in a half-cylinder with periodic coeffi- cients on all variables except for one was considered, the exponential stabilization