Let \(\mathcal {M}\) be a smooth, closed and connected manifold of dimension \(n\in \mathbb {N}\) , endowed with a Riemannian metric g. Moreover, let \(\mathcal {B}\) be an \((n+1)\) -dimensional compact manifold with boundary equal to \(\mathcal {M}\) . Endow \(\mathcal {B}\) with a Riemannian metric h such that, in local coordinates \((x,y)\in [0,1)\times \mathcal {M}\) on the collar part of the boundary, it admits the warped product form \(h=dx^{2}+x^{2}g(y)\) . We consider the homogeneous heat equation on \((\mathcal {B},h)\) and find an arbitrary long asymptotic expansion of the solutions with respect to x near 0. It turns out that the spectrum of the Laplacian on \((\mathcal {M},g)\) determines explicitly the above asymptotic expansion and vice versa.
