We studied the heat conduction and the mean curvature flow in the whole space. In the initial problem for the heat equation in the whole space, the solution does not necessarily converge to any fixed values as time goes to infinity, even if we consider bounded initial values. Recently it is shown that the same phenomenon occurs in the curvature flow problem and, more precisely, that the large time behavior of solutions of these two equations are equivalent under some assumptions on the initial value. To understand this fact, we introduced a technique for analyzing parabolic equations as the heat equation with some heat flux.

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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.

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