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

Abstract: 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}-\Delta u = \mu & \text{in}\ (0,T)\times\Omega,\[0.7 ex] u(0,x)=u_0 & \text{in}\ \Omega, \end{cases} $$ where $\mu$ is a general, possibly singular, Radon measure which does not depend on time, and $u_0\in L^{1}(\Omega)$. 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.