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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 $t \in (0,1)$.

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Zen, I am going to betray my immense ignorance of non-soft analysis here, and ask: if your function $u$ is a priori only continuous, in what sense are the partial derivatives in the heat equation being interpreted? Are these distributional (partial) derivatives? –  Yemon Choi Feb 22 '11 at 8:59
A very good question, Yemon: I don't know what is the best interpretation, since that might depend on the answer! But assuming them to be $C^\infty$ probably doesn't make the question significantly easier (but I might be wrong). I think we even have a kind of elliptic regularity, so that distributional derivatives implies $C^\infty$ classical partial derivatives (and we might even, if we're very lucky, get this just by assuming pointwise derivatives exist) - but I might be very wrong! –  Zen Harper Feb 22 '11 at 9:30
Is this maybe what you are looking for: A. Tychonoff, Théorème d’unicité pour l’équation de la chaleur Math. Sbornik 42 (1935), 199–215. The text is in French but not so difficult to read. The fulltext available for free: mathnet.ru/links/5e8a21739eefe545e74e49e2d1f30996/sm6410.pdf –  user10893 Apr 9 '11 at 11:11
Thanks very much, CJ, for that link. I'm having computer problems right now, but I'll look at it as soon as this evil metal monster allows me to. –  Zen Harper Apr 13 '11 at 9:11
Thanks again, CJ. Although that paper is interesting, it only seems to consider bounds on $|u(x,t)|$ depending on $x$, but independent of $t$, i.e. $|u(x,t)| \leq V(x)$ for all $t>0$ and some $V$; whereas I am looking for bounds $|u(x,t)| \leq W(t)$ depending on $t$, not $x$. [Although it's possible my understanding of French is too poor]. I shall re-edit my question to try to make it more clear. –  Zen Harper Apr 14 '11 at 7:53
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