Stabilization of the solution of the Cauchy problem for parabolic equations (Q1076230)

The authors consider a linear uniformly parabolic operator of second order with symmetric and bounded coefficients. The behavior of the solution u(x,t) of the Cauchy problem as \(t\to \infty\) is investigated. In particular they consider the pointwise stabilization problem, i.e. the problem of the existence of the limit \(\lim_{t\to \infty}u(x,t)=\ell (x),\) and the problem of the existence of the mean of u(x,t) with respect to time, i.e. the existence of the limit \(\lim_{t\to \infty}t^{- 1}\int^{t}_{0}u(x,s)ds=\ell (x).\) Under some averaging conditions on the operator, the authors give necessary and sufficient conditions on the initial data for the resolution of the above problems.