Asymptotic behavior of solutions for linear parabolic equations with general measure data

In this Note we deal with the asymptotic behavior as t tends to infinity of solutions for linear parabolic equations whose model is {ut - Delta u = in (0, T) x Omega, u(0,x) = u(0) in Omega, where mu is a general, possibly singular, Radon measure which does not depend on time, and u(0) is an element of L-1 (Omega). We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by Stampacchia (1965)) of the associated elliptic problem.