Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition A null temperature function is a continuous function $u = u(x,t) : \mathbb{R} \times [0, \infty) \to \mathbb{R}$ such that the heat equation is satisfied on the interior, i.e. \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \quad \text{ for } t>0, \] but $u(x, 0) \equiv 0$ for all $x \in \mathbb{R}$.
EDIT: if you wish, assume all partial derivatives (of all orders) to exist and be continuous also [in the interior]; I expect this doesn't actually make much difference, by elliptic regularity arguments.
Theorem There exists a null temperature function satisfying $ |u(x,t)| < \exp(A/t)$ with $A>0$, such that $u(x,t) \not\equiv 0$ for some $t>0$.
Theorem Let $u$ be a null temperature function satisfying $|u(x,t)| \leq A \exp(B t^{-\delta})$, for some $A,B>0$ and $\delta<1$. Then $u \equiv 0$.
[I should mention that these are not my theorems! References:
S.-Y. Chung, D. Kim. An example of nonuniqueness of the Cauchy problem for the heat equation. Comm. Partial Differential Equations 19 (1994), no. 7-8, 1257–1261. MR1284810 (95c:35114)
S.-Y. Chung. Uniqueness in the Cauchy problem for the heat equation. Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 3, 455–468. MR1721765 (2000h:35060)
Also discussed briefly, with connections to the uniqueness problem for the Laplace transform, in my paper "Laplace transform representations and Paley–Wiener theorems for functions on vertical strips"]
Vague Questions Besides the results above, what is known about the class of null temperature functions? Clearly it is a vector space; can it be given a "natural" Banach space norm? Can we represent it (or nice subspaces of it) in any nice way? What kind of growth rates are possible?
EDIT Precise question - maximal growth rates Is there some universal function $\varphi : (0,1) \to \mathbb{R}$ with the following property?
For every non-trivial Null Temperature Function $u$ such that $M(t) = \sup_x |u(x,t)| < \infty$ for each $t>0$, there is some $C < \infty$ such that $M(t) \leq C \varphi(t)$ for all $t \in (0,1)$.
