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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
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