Geodesic
An investigation of the behavior of $L_2$-norms of the solutions of strongly parabolic systems with dissipation
Izvestiya. Mathematics , Tome 7 (1973) no. 3, pp. 675-689.

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It is established under certain interrelations between the growth of the initial function and the dissipation intensity that an increasing solution becomes with the passage of time a bounded and ultimately decreasing function of the spatial coordinates. The behavior of $L_2$-norms (containing weights depending in a special manner on the time $t$) of the solutions is investigated for this purpose. The results are essentially based on inequalities established in the paper between the norms and quasinorms introduced and which, in our view, are of independent interest. 
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S. D. Èidel'man; F. O. Porper. An investigation of the behavior of $L_2$-norms of the solutions of strongly parabolic systems with dissipation. Izvestiya. Mathematics , Tome 7 (1973) no. 3, pp. 675-689. http://geodesic.mathdoc.fr/item/IM2_1973_7_3_a11/

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