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On the stabilization of solutions of parabolic equations

V. V. Zhikov


Abstract: 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.

UDC: 517.946

MSC: Primary 35K15, 35B40; Secondary 35K30

Received: 22.11.1976

 English version: Mathematics of the USSR-Sbornik, 1977, 33:4, 519–537

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