ON UNIFORM STABILIZATION OF SOLUTIONS OF THE FIRST MIXED PROBLEM FOR A PARABOLIC EQUATION

The first mixed problem with a homogeneous boundary condition is considered for a linear parabolic equation of second order. It is assumed that the unbounded domain satisfies the following condition: there exists a positive constant such that for any point of the boundary For a certain class of initial functions , which includes all bounded functions, the following condition is a necessary and sufficient condition for uniform stabilization of the solution to zero: The proof of the stabilization condition is based on an estimate of the Green function that takes account of its decay near the boundary.