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Stability of the static spike autosolitons in the Gray-Scott model. (English) Zbl 1012.35042

An asymptotic linear stability analysis of the static spike autosolitions (ASs) in the Gray-Scott model of an autocatalytic chemical reaction is performed. The authors found that in one dimension these ASs destabilize with respect to pulsations the onset of traveling motion when the inhibitor is slow. In higher dimensions, the one-dimensional static spike ASs is always unstable with respect to corrugation and wriggling. The higher-dimensional radially symmetric static spike ASs may destabilize with respect to the radially nonsymmetrical fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow.

MSC:

35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
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