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

Optimal existence classes and nonlinear--like dynamics in the linear heat equation in ${\mathbb R}^d$