Movement of hot spots over unbounded domains in \({\mathbb{R}}^ N\) (Q1323922)

The authors study the heat equation a) on a half space \(\mathbb{R}^+_n\), b) on the exterior of a ball with radial initial data, c) on the exterior of a bounded set \(\Omega\), and investigate the location and movement of hot spots for large \(t\) and under boundary conditions of Dirichlet, Neumann and Robin type. In the half plane case and for Neumann boundary conditions, for instance, the hot spot moves to the boundary in finite time, which (partially) answers a conjecture of J. Rauch. This and other results are shown by the moving plane method. In case b) one has \(r\) as a single space variable and one can use lap-number arguments to show that the set of hot spots becomes a single sphere after finite time. For the situation c) the results are less specific.