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

*(English)*Zbl 1124.35318Summary: In this note we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is

$$\left\{\begin{array}{cc}{u}_{t}-{\Delta}u=\mu \hfill & \text{in}\phantom{\rule{4.pt}{0ex}}(0,T)\times {\Omega},\hfill \\ u(0,x)={u}_{0}\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega},\hfill \end{array}\right.$$

where $\mu $ is a general, possibly singular, Radon measure which does not depend on time, and ${u}_{0}\in {L}^{1}\left({\Omega}\right)$. 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.

##### MSC:

35K15 | Second order parabolic equations, initial value problems |

35R05 | PDEs with discontinuous coefficients or data |

35B40 | Asymptotic behavior of solutions of PDE |