By M Z Zhurovsʹkyĭ; et al

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**Additional resources for Evolution inclusions and variation inequalities for earth data processing. / III, Long-time behavior of evolution inclusions solutions in earth data analysis**

**Sample text**

T; / W X ! t1 ; / be compact. t1 C t; / is also compact for any t 2 C . Proof. X /. t1 ; B//. t1 ; B// is compact in X (see [2, p. 42]). t C t1 ; B// is precompact. 2. t0 ; / W X ! t; / W X ! 1) where m2 W RC ! R is a continuous map and m1 W C ! t/ ! 0, as t ! 1. Then G is asymptotically upper semicompact. dist. Proof. X /. tn ; B/, tn ! 1, and prove that N is completely bounded. For a fixed > 0, we take t1 D t1 . B/. B//. t0 ; 9 Á as well. B/. So, N is completely bounded and by Hausdorff’s theorem it is precompact.

The third method is based on the idea of replacing the phase space of the problem by a specially constructed space of trajectories and considering on it the single-valued shift semigroup [22, 23, 55, 74]. The notion of trajectory attractor, that is, the global attractor (defined in the classical way) of the shift semigroup, also appeared to be very useful for the investigation of nonlinear evolution equations without uniqueness. , [22, 23, 41, 56, 64, 67] and also [26]). As pointed out in [9], the disadvantage of this method is that the direct connection with the natural phase space of the problem is lost.

Tolstonogov AA (1992) Solutions of evolution inclusions I. Siberian Math J. 1007/ BF00970899 233. Tolstonogov AA, Umanskii YuA (1992) On solutions of evolution inclusions II. Siberian Math J. 1007/BF00971135 234. Valero J (2001) Attractors of parabolic equations without uniqueness. J Dyn Differ Equat. 1023/A:1016642525800 235. Valero J, Kapustyan AV (2006) On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems. J Math Anal Appl. 042 236. Vishik M, Chepyzhov VV (2002) Trajectory and global attractors of three-dimensional Navier–Stokes systems.