Dynamics of shadow system of a singular Gierer-Meinhardt system on an evolving domain
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University of Chester; Polytechnic Institute of Setubal; University of Lisbon; Sussex University
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The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer-Meinhardt model on an isotropically evolving domain. 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 is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter 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. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.Citation
Kavallaris, N., Bareira, R., & Madzvamuse, A. (2020 - awaiting publication). Dynamics of shadow system of a singular Gierer-Meinhardt system on an evolving domain. Journal of Nonlinear Science, 31(5). https://doi.org/10.1007/s00332-020-09664-3Publisher
SpringerJournal
Journal of Nonlinear ScienceAdditional Links
http://link.springer.com/journal/332Type
ArticleDescription
The final publication is available at Springer via DOI TBCEISSN
1432-1467ae974a485f413a2113503eed53cd6c53
10.1007/s00332-020-09664-3
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