On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system

Hdl Handle:
http://hdl.handle.net/10034/620434
Title:
On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system
Authors:
Kavallaris, Nikos I.; Suzuki, Takashi
Abstract:
The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.
Affiliation:
University of Chester; Osaka University
Citation:
Kavallaris, N. I., & Suzuki, T. (2017). On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system. Nonlinearity, 30(5), 1734-1761. https://doi.org/10.1088/1361-6544/aa64b2
Publisher:
London Mathematical Society
Journal:
Nonlinearity
Publication Date:
21-Mar-2017
URI:
http://hdl.handle.net/10034/620434
DOI:
10.1088/1361-6544/aa64b2
Additional Links:
http://iopscience.iop.org/article/10.1088/1361-6544/aa64b2/meta
Type:
Article
Language:
en
Description:
This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1361-6544/aa64b2/meta
EISSN:
1361-6544
Appears in Collections:
Mathematics

Full metadata record

DC FieldValue Language
dc.contributor.authorKavallaris, Nikos I.en
dc.contributor.authorSuzuki, Takashien
dc.date.accessioned2017-03-14T09:57:02Z-
dc.date.available2017-03-14T09:57:02Z-
dc.date.issued2017-03-21-
dc.identifier.citationKavallaris, N. I., & Suzuki, T. (2017). On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system. Nonlinearity, 30(5), 1734-1761. https://doi.org/10.1088/1361-6544/aa64b2en
dc.identifier.doi10.1088/1361-6544/aa64b2-
dc.identifier.urihttp://hdl.handle.net/10034/620434-
dc.descriptionThis is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1361-6544/aa64b2/metaen
dc.description.abstractThe purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.en
dc.language.isoenen
dc.publisherLondon Mathematical Societyen
dc.relation.urlhttp://iopscience.iop.org/article/10.1088/1361-6544/aa64b2/metaen
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/en
dc.subjectPattern formationen
dc.subjectTuring instabilityen
dc.subjectActivator-inhibitor systemen
dc.subjectShadow-systemen
dc.subjectInvariant regionsen
dc.subjectDiffusion-driven blow-upen
dc.titleOn the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt systemen
dc.typeArticleen
dc.identifier.eissn1361-6544-
dc.contributor.departmentUniversity of Chester; Osaka Universityen
dc.identifier.journalNonlinearityen
dc.date.accepted2017-03-06-
or.grant.openaccessYesen
rioxxterms.funderUnfundeden
rioxxterms.identifier.projectUnfundeden
rioxxterms.versionAMen
rioxxterms.licenseref.startdate2217-03-14-
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