  We present a new connection between the classical theory of full and
truncated moment problems and the theory of partial differential equations, as
follows. For the classical heat equation $\partial_t u = \nu \Delta u$, with
initial data $u_0 \in\mathcal{S}(\mathbb{R}^n)$, we first compute the moments
$s_{\alpha}(t)$ of the unique solution $u \in \mathcal{S}(\mathbb{R}^n)$. These
moments are polynomials in the time variable, of degree comparable to $\alpha$,
and with coefficients satisfying a recursive relation. This allows us to define
the polynomials for any sequence, and prove that they preserve some of the
features of the heat kernel. In the case of moment sequences, the polynomials
trace a curve (which we call the heat curve) which remains in the moment cone
for positive time, but may wander outside the moment cone for negative time.
This provides a description of the boundary points of the moment cone which are
also moment sequences. \ We also study how the determinacy of a moment sequence
behaves along the heat curve. Next, we consider the transport equation
$\partial_t u = ax \cdot \nabla u$, and conduct a similar analysis. Along the
way we incorporate several illustrating examples.
