Point-condensation for a reaction-diffusion system
This paper deals with a system of the form \[ (*)\quad d_ 1u''+f(u,v)=0,\quad d_ 2v''g(u,v)=0\quad on\quad [0,1] \] subject to homogeneous Neumann boundary conditions at 0 and 1. The diffusion coefficients \(d_ 1\) and \(d_ 2\) are positive parameters, here. The problem arises as a normalized steady state system from a reaction- diffusion system proposed by Gierer and Meinhardt. Assuming \(d_ 2\) to be sufficiently large the author shows that the system possesses ``point-condensing'' solutions, if \(d_ 1\) is sufficiently small. Moreover, he obtains the following global result: Depending on whether bifurcation from the constant solutions occurs or not, there exists a continuum either connecting them to the point- condensing solutions or connecting the point-condensing solutions to another family of such solutions.
scientific article; zbMATH DE number 4019595
scientific article