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 Mat. Sb., 2007, Volume 198, Number 11, Pages 67–106 (Mi msb3792)

The problem of birth of autowaves in parabolic systems with small diffusion

A. Yu. Kolesova, N. Kh. Rozovb, V. A. Sadovnichiib

a P. G. Demidov Yaroslavl State University
b M. V. Lomonosov Moscow State University

Abstract: A parabolic reaction-diffusion system with zero Neumann boundary conditions at the end-points of a finite interval is considered under the following basic assumptions. First, the matrix diffusion coefficient in the system is proportional to a small parameter $\varepsilon>0$, and the system itself possesses a spatially homogeneous cycle (independent of the space variable) of amplitude of order $\sqrt\varepsilon$ born by a zero equilibrium at an Andronov–Hopf bifurcation. Second, it is assumed that the matrix diffusion depends on an additional small parameter $\mu\ge0$, and for $\mu=0$ there occurs in the stability problem for the homogeneous cycle the critical case of characteristic multiplier 1 of multiplicity 2 without Jordan block. Under these constraints and for independently varied parameters $\varepsilon$ and $\mu$ the problem of the existence and the stability of spatially inhomogeneous auto-oscillations branching from the homogeneous cycle is analysed.
Bibliography: 16 titles.

DOI: https://doi.org/10.4213/sm3792

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English version:
Sbornik: Mathematics, 2007, 198:11, 1599–1636

Bibliographic databases:

UDC: 517.957
MSC: 35K57, 35B10, 35B32

Citation: A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “The problem of birth of autowaves in parabolic systems with small diffusion”, Mat. Sb., 198:11 (2007), 67–106; Sb. Math., 198:11 (2007), 1599–1636

Citation in format AMSBIB
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