  We analyse the behaviour of solutions of the linear heat equation in
${\mathbb R}^d$ for initial data in the classes $M_\varepsilon({\mathbb R}^d)$
of Radon measures with $\int_{{\mathbb R}^d}{\rm e}^{-\varepsilon|x|^2}\,{\rm
d}|u_0|<\infty$. We show that these classes are in some sense optimal for local
and global existence of non-negative solutions: in particular $M_0({\mathbb
R}^d)=\cap_{\varepsilon>0}M_\varepsilon({\mathbb R}^d)$ consists precisely of
those initial data for which the a solution of the heat equation can be given
for all time using the heat kernel representation formula. After considering
properties of existence, uniqueness, and regularity for such initial data,
which can grow rapidly at infinity, we go on to show that they give rise to
properties associated more often with nonlinear models. We demonstrate the
finite-time blowup of solutions, showing that the set of blowup points is the
complement of a convex set, and that given any closed convex set there is an
initial condition whose solutions remain bounded precisely on this set at the
`blowup time'. We also show that wild oscillations are possible from
non-negative initial data as $t\to\infty$ (in fact we show that this behaviour
is generic), and that one can prescribe the behaviour of $u(0,t)$ to be any
real-analytic function $\gamma(t)$ on $[0,\infty)$.
