# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $O\left(1\right)$ time scale.
##### MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 34B15 Nonlinear boundary value problems for ODE 34E15 Asymptotic singular perturbations, general theory (ODE) 35B40 Asymptotic behavior of solutions of PDE 35K57 Reaction-diffusion equations 92C15 Developmental biology, pattern formation