Necessary and Sufficient Conditions of Stabilization of Solutions of the First Boundary-Value Problem for a Parabolic Equation

Necessary and sufficient conditions are established for the stabilization of the solution of the first boundary-value problem for a divergent second-order parabolic equation with no lower-order terms. For an arbitrary (possibly unbounded) domain Q ? ? ^N , N?≥?3, with an initial function u ₀(x) defined in Q for t?=?0, one studies the influence of Q on the stabilization of the solution in the cylinder Z?=?Q?×?0,∞) with the homogeneous boundary conditions on the lateral surface of Z.