Point condensation generated by a reaction-diffusion system in axially symmetric domains
type: Item
instance of: scholarly article
title: Point condensation generated by a reaction-diffusion system in axially symmetric domains
zbMATH Open document ID: 0843.35006
DOI: 10.1007/BF03167294
author: Wei-Ming Ni
author: Izumi Takagi
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 1995-12-03T00:00:00Z
review text: The authors study the stationary problem for a reaction-diffusion system of activator-inhibitor type, which models biological pattern formation, in an axially symmetric domain. The system has the form \[ \begin{aligned} {\partial a\over \partial t} & = D_a \Delta a- N_a a+ \rho_a \Biggl(C_a{a^p\over h^q}+ \rho_0\Biggr),\\ {\partial h\over \partial t} & = D_h \Delta h- N_h h+ C_h \rho_h {a^r\over h^s}.\end{aligned} \] Here \(a= a(x, t)\), \(h= h(x, t)\) are unknown functions, \(D_a, D_h\), \(N_a, C_a, C_h\), \(\rho_a\), \(\rho_h\) are all positive constants, while \(\rho_0\) is a nonnegative constant; \(\Delta= \sum^N_{j= 1} \partial^2/\partial x^2_j\). The exponents \(p\), \(q\), \(r\), \(s\) are assumed to satisfy the conditions \(p> 1\), \(q> 0\), \(r> 0\), \(s\geq 0\) and \(0< (p- 1)/q< r/(s+ 1)\). It is shown that the system has multi-peak stationary solutions such that the activator \(a\) is localized around some boundary points if the activator diffuses very slowly and the inhibitor \(h\) diffuses rapidly enough.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 812644
zbMATH Keywords: semilinear Neumann problem
zbMATH Keywords: multi-peak stationary solutions
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 812644