Metastability in the Shadow System for Gierer-Meinhardt's Equations

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dc.contributor.authorde Groen, Pieter P. N.
dc.contributor.authorKaradzhov, Georgi E.
dc.date.accessioned2020-08-10T19:48:45Z
dc.date.available2020-08-10T19:48:45Z
dc.date.issued2002-06-02
dc.description.abstractIn this paper we study the stability of the single internal spike solution of the shadow system for the Gierer-Meinhardt equations in one space dimension. It is well-known, that the linearization around this spike consists of a differential operator plus a non-local term. For parameter values in certain subsets of the 3D (p, q, r)-parameter space we prove that the non-local term moves the negative O(1) eigenvalue of the differential operator to the positive (stable) half plane and that an exponentially small eigenvalue remains in the negative half plane, indicating a marginal instability (dubbed "metastability"). We also show, that for parameters (p, q, r) in another region, the O(1) eigenvalue remains in the negative half plane. In all asymptotic approximations we compute rigorous bounds for the order of the error.
dc.description.departmentMathematics
dc.formatText
dc.format.extent22 pages
dc.format.medium1 file (.pdf)
dc.identifier.citationde Groen, P., & Karadzhov, G. (2002). Metastability in the shadow system for Gierer-Meinhardt's equations. <i>Electronic Journal of Differential Equations, 2002</i>(50), pp. 1-22.
dc.identifier.issn1072-6691
dc.identifier.urihttps://hdl.handle.net/10877/12349
dc.language.isoen
dc.publisherSouthwest Texas State University, Department of Mathematics
dc.rightsAttribution 4.0 International
dc.rights.holderThis work is licensed under a Creative Commons Attribution 4.0 International License.
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.sourceElectronic Journal of Differential Equations, 2002, San Marcos, Texas: Southwest Texas State University and University of North Texas.
dc.subjectSpike solution
dc.subjectSingular perturbations
dc.subjectReaction-diffusion equations
dc.subjectGierer-Meinhardt equations
dc.titleMetastability in the Shadow System for Gierer-Meinhardt's Equations
dc.typeArticle

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