Geodesic
Justification of the method of quasinormal forms for Hutchinson's equation with a~small diffusion coefficient
Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 749-768.

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Although we consider only the concrete problem indicated in the title (the proofs of general theorems would be bulky), our arguments can be adapted for a wide class of singularly perturbed systems of reaction-diffusion type with time-delay in the ordinary part. 
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Yu. S. Kolesov. Justification of the method of quasinormal forms for Hutchinson's equation with a~small diffusion coefficient. Izvestiya. Mathematics , Tome 65 (2001) no. 4, pp. 749-768. http://geodesic.mathdoc.fr/item/IM2_2001_65_4_a6/

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