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The existence and stability of asymmetric spike patterns for the Schnakenberg model. (English) Zbl 1152.35397
Summary: Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction-diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields k 1 spikes that have a common small amplitude and k 2 =k-k 1 spikes that have a common large amplitude. A k-spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction-diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O(1) time scale.
MSC:
35K50Systems of parabolic equations, boundary value problems (MSC2000)
34B15Nonlinear boundary value problems for ODE
34E15Asymptotic singular perturbations, general theory (ODE)
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations
92C15Developmental biology, pattern formation