Geodesic
On the stabilization of solutions of weakly coupled cooperative parabolic systems
Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 177-192

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In this work it is shown that almost all solutions of weakly coupled parabolic systems either stabilize or blow up. It is also shown, under an additional condition, that almost all solutions either blow up or become componentwise monotonic, starting from some instant. 
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M. P. Vishnevskii. On the stabilization of solutions of weakly coupled cooperative parabolic systems. Sbornik. Mathematics, Tome 77 (1994) no. 1, pp. 177-192. http://geodesic.mathdoc.fr/item/SM_1994_77_1_a10/