Kolesov, A. Yu; Rozov, N. Kh; Sadovnichiĭ, V. A. The problem of birth of autowaves in parabolic systems with small diffusion. (English. Russian original) Zbl 1160.35450 Sb. Math. 198, No. 11, 1599-1636 (2007); translation from Mat. Sb. 198, No. 11, 67-106 (2007). Summary: 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 \( \mu\geqslant 0\), 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. MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K57 Reaction-diffusion equations 35B10 Periodic solutions to PDEs 35B32 Bifurcations in context of PDEs 35K55 Nonlinear parabolic equations 35B25 Singular perturbations in context of PDEs Keywords:zero Neumann boundary conditions; Andronov-Hopf bifurcation PDFBibTeX XMLCite \textit{A. Y. Kolesov} et al., Sb. Math. 198, No. 11, 1599--1636 (2007; Zbl 1160.35450); translation from Mat. Sb. 198, No. 11, 67--106 (2007) Full Text: DOI