Summary: In this note we deal with the asymptotic behavior as tends to infinity of solutions for linear parabolic equations whose model is
where is a general, possibly singular, Radon measure which does not depend on time, and . We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by G. Stampacchia [Ann. Inst. Fourier 15, No. 1, 189–257 (1965) and Colloques Int. Centre nat. Rech. Sci. 146, 189–258 (1965; Zbl 0151.15401)]) of the associated elliptic problem.