We study the asymptotic behaviour of solutions to the Cauchy problem for linear degenerate parabolic equations. The coefficients of the equations may grow or tend to zero as $|x|\rightarrow \infty$. We establish necessary and sufficient conditions on the initial data which guarantee the stabilization of the solution to a constant.
"On stabilization of solutions of the Cauchy problem for linear degenerate parabolic equations." Adv. Differential Equations 14 (7/8) 621 - 641, July/August 2009.