Geodesic
On~the stabilization of solutions of parabolic equations
Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 519-537

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In this paper, a method for studying stabilization properties of second order, and also higher order, equations is presented. In particular, pointwise stabilization criteria are obtained for the equation $cu_t=\Delta u$, the only requirement on the coefficient $c(x)$ being the existence of a mean value. This result generalizes known results of Gushchin and Mihailov, as well as the corresponding results for the equation of heat conduction. Analogous criteria are developed for the equation $cu_t+(-1)^m\Delta^mu=0$. Stabilization criteria are proved for other equations as well. Bibliography: 8 titles. 
@article{SM_1977_33_4_a4,
     author = {V. V. Zhikov},
     title = {On~the stabilization of solutions of parabolic equations},
     journal = {Sbornik. Mathematics},
     pages = {519--537},
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     volume = {33},
     number = {4},
     year = {1977},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1977_33_4_a4/}
}
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V. V. Zhikov. On~the stabilization of solutions of parabolic equations. Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 519-537. http://geodesic.mathdoc.fr/item/SM_1977_33_4_a4/