Stationary circular target patterns in a surface burning reaction

A simple model for burning on the circular face of a substrate is analyzed. It is shown that spatial patterns can form, in which the temperature develops hot and cold regions arranged in concentric circular rings. A linearized study shows the parameter values for which small amplitude patterns are stable. The fully non-linear equations are then solved using an efficient shooting method in the spatial variable, and an extremely complicated bifurcation diagram is obtained, from which it follows that multiple solutions occur at the same values of the defining parameters. The effect of heat leakage at the edges of the circular region is considered, and complicated non-linear behaviour occurs in this case also. Seven different temperature patterns, all co-existing at the same parameter values, are presented in a particular instance.