  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.
