Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system

This paper is concerned with stationary solutions of a reaction-diffusion- advection system arising in surface chemistry. Hildebrand et al (2003 New J. Phys. 5 61) have constructed stationary stripe (or spot) solutions of the system in the singular perturbation case and shown a numerical result that the set of stripe (or spot) solutions forms a saddle-node bifurcation curve with respect to a diffusion coefficient. In this paper, we introduce a shadow system in the limiting case that another diffusion and an advection coefficient tend to infinity. Furthermore we obtain the bifurcation structure of stationary solutions of the shadow systems in the one-dimensional case. This structure involves saddle-node bifurcation curves which support the above numerical result in Hildebrand et al (2003 New J. Phys. 5 61, figure 9). Our proof is based on the combination of the bifurcation, the singular perturbation and a level set analysis.