The first part of the talk focuses on the study of the behaviour at infinity of solutions to a second order elliptic equation with first order terms stated in a half-cylinder. The coefficients of the equation are assumed to be measurable and bounded; Neumann boundary condition is imposed on the lateral boundary of the cylinder, while on the base we assign the Dirichlet boundary condition. Under some natural assumptions we study the existence of a bounded solution and its stabilization to a constant at the exponential rate. Also we provide a necessary and sufficient condition for the uniqueness of a bounded solution. The second part of the talk is devoted to the study of a convection-diffusion operator in an infinite cylinder being a union of two nonintersecting half-cylinders with a junction at the origin. The coefficients of the equation are supposed to be periodic in each of these cylinders, and the Neumann boundary condition is imposed on the lateral boundary of the cylinder. The existence of a bounded solution and its qualitative properties are discussed. As an application of the obtained results we consider the homogenization problem of a stationary convection-diffusion equation in a thin cylinder being a union of two nonintersecting rods with a junction at the origin.