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

A number of properties of the Green function of the second mixed problem for a parabolic equation of second order on ( an arbitrary domain in ) are established. By means of these results a criterion is proved for the uniform stabilization of a solution: the existence of a uniform limit of the spherical mean of the initial function (extended by zero outside ) is necessary and sufficient for the uniform stabilization of a solution of the problem considered, with a bounded initial function under a certain condition on the unbounded domain .The basic properties of the Green function are obtained on the basis of an estimate of the solution of the problem with a compactly supported initial function in terms of the norm of the initial function in .Bibliography: 53 titles.