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Asymptotic behavior of solutions for linear parabolic equations with general measure data. (English) Zbl 1124.35318

Summary: 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}}\left(0,T\right)×{\Omega },\hfill \\ u\left(0,x\right)={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