Point condensation generated by a reaction-diffusion system in axially symmetric domains
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.
scientific article; zbMATH DE number 812644
scientific article