Hot spots on cones and warped product manifolds

We study extrema of solutions to the heat equation (i.e. hot spots) on a class of warped product manifolds of the form $([0,L]\times M,dr^2+f(r)^2h)$ where $(M,h)$ is a closed Riemannian manifold. We prove that, under certain conditions on the warping function $f$, the statement of Rauch's hot spots conjecture holds for the corresponding warped product. We then go on to study the long-time behavior of hot spots on infinite cones over closed Riemannian manifolds. In this case, under appropriate hypotheses on the initial condition, there are four possible long-time behaviors depending only on the spectral gap of the fiber $(M,h)$.