Abstract
We will mathematically describe the stable steady states of the shadow system of the activator-inhibitor type in a disk, using the number and the locations of the critical points of the function. Specifically, if the steady state is stable, then the solution has exactly two critical points on the disk and they are on the boundary. Hence the shape of the stable steady state is like a boundary one-spike layer. We will see that our problem can be reduced to the nonlinear "hot spots" conjecture, and that this conjecture is a fundamental theorem in studying the shape of the stable pattern of the shadow system.
