Geodesic
The stabilization of solutions of the neralized Cauchy problem for ultraparabolic equations
Izvestiya. Mathematics , Tome 3 (1969) no. 2, pp. 345-355.

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By using an “integral” representation of the solution of the generalized Cauchy problem for ultraparabolic equations, a necessary and sufficient condition for the stabilization of the solution has been obtained for the class of positive initial functionals. In the class of “bounded with respect to translation” functionals, it has been proved that a necessary and sufficient condition for the stabilization of the solution in a weak sense is the existence of a generalized spherical limiting mean. 
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Yu. N. Drozhzhinov. The stabilization of solutions of the neralized Cauchy problem for ultraparabolic equations. Izvestiya. Mathematics , Tome 3 (1969) no. 2, pp. 345-355. http://geodesic.mathdoc.fr/item/IM2_1969_3_2_a2/

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