A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems
This paper concerns the reaction-diffusion system \[ \left.\begin{aligned} u_t= \varepsilon^2u_{xx}&+ f(u,v)\\ v_t=Dv_{xx}&+ g(u,v)\end{aligned}\right\}\quad 0<x<1,\;t>0 \] with homogeneous Neumann boundary conditions and nonlinearities of propagator-controller type. The aim is existence and stability of a family (parametrized by small \(\varepsilon>0)\) of equilibria with one internal transition layer. The main techniques are: construction of approximate solutions with high degree of accuracy, estimation of the principal eigenvalue of the linearization around the approximate solution (SLEP method), Lyapunov-Schmidt reduction and two applications of the implicit function theorem.
scientific article; zbMATH DE number 5011027