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

Abstract

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.