STABILIZATION OF SOLUTIONS OF THE FIRST MIXED PROBLEM FOR A PARABOLIC EQUATION OF SECOND ORDER

The behavior for large time of the solution in an unbounded domain of the first mixed problem for the parabolic equation (1) (2)with initial function , , , is investigated. It is shown that the function , which for each fixed is the first eigenvalue of the Dirichlet problem for the operator in , for a certain class of domains determines the rate at which the solution tends to zero as . Namely, let be the function inverse to the monotone increasing function . Then for all and all in (3)Here the constant depends only on and of (2), while and depend on , , and . It is proved that for a certain class of domains the estimate (3) is in a sense best possible.Bibliography: 13 titles.