Construction and stability analysis of transition layer solutions in reaction-diffusion systems
The problem here considered is the construction and the stability analysis of transition layers associated to the system \[ u_ t=\epsilon^ 2u_{xx}+f(u,v),\quad 0<x<1,\quad t>0 \] \[ v_ t=\sigma^{-1}v_{xx}+g(u,v),\quad 0<x<1,\quad t>0 \] \[ u_ x=v_ x=0\quad at\quad x=0,1 \] (\(\epsilon\),\(\sigma\) being small positive parameters). Under some specific assumptions on the source terms f,g a careful analysis is performed about the existence and the behaviour of the equilibrium solutions as \(\epsilon\),\(\sigma\) approach zero, showing the occurrence of a transition boundary layer for u. In particular a result about the stability of such solutions is obtained by means of the spectral analysis of the operator which linearizes \(\epsilon^ 2u''+f(u,v)\), \(\sigma 'v''+g(u,v)\) with b.c. \(u'=v'=0\) at \(x=0,1\).
scientific article; zbMATH DE number 4163342
scientific article