On the line of earlier works of the first author, the paper is concerned with the uniform stabilization of the solution of the second mixed problem for a second order linear parabolic equation. The authors prove the stabilization criteria for large classes of unbounded domains including plane domains such as \[ \{(x_ 1,x_ 2): x_ 1>0,\quad | x_ 2| <kx\quad a_ 1,\quad k=const.>0,\quad a\in (0,1)\}. \] One also obtains estimates for the corresponding Green’s function.