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Stability of least energy patterns of the shadow system for an activator-inhibitor model. (English) Zbl 1200.35172
Summary: Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal “energy” is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.

MSC:
35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35B35 Stability in context of PDEs
92C15 Developmental biology, pattern formation
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