We consider the Cauchy-Dirichlet problem of the heat equation in the exterior domain of a ball, and study the movement of hot spots $H(t)$ as $t\to\infty$. In particular, we give a rate for the hot spots to run away from the boundary of the domain as $t\to\infty$. Furthermore we give a sufficient condition for the hot spots to consist of only one point after a finite time.
"Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition." Adv. Differential Equations 12 (10) 1135 - 1166, 2007.