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

In this paper we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is $$ \begin{cases} u_{t}-Δu = μ& \text{in}\ (0,T)\timesΩ,\\[0.7 ex] u(0,x)=u_0 & \text{in}\ Ω, \end{cases} $$ where $μ$ is a general, possibly singular, Radon measure which does not depend on time, and $u_0\in L^{1}(Ω)$. We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by G. Stampacchia) of the associated elliptic problem.