"Wild" solutions of the heat equation: how to graph them? It has long been known that the Cauchy initial-value problem for the
classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't
have unique solutions, without additional assumptions.  In particular:

Theorem. There exist nontrivial $C^\infty$ functions $u :
    (-\epsilon, \epsilon) \times \mathbb{R} \to \mathbb{R}$ satisfying
      the heat equation
  $$\partial_t u - \partial_x^2 u = 0$$
  with the initial condition $u(0,x) = 0$.

These functions must somehow represent disturbances arriving from
infinity in finite time.  They seem pretty weird to me and I think a
picture would help with intuition.

I would like to see a graph of such a function.  Where can I find one,
  or how can I generate it?

There are some explicit examples known.  For instance, Rosenbloom and
Widder [1] give the following:
$$u(t,x) = \int_0^\infty e^{-y^{4/3}} y \cos(\sqrt{3} y^{4/3}) 
(e^{xy} \cos(xy + 2ty^2) + e^{-xy} \cos(xy-2ty^2))\,dy$$
as well as
$$v(t,x) = \int_{a-i\infty}^{a+i\infty} e^{st + x\sqrt{s} - s^{2/3}}\,ds.$$
Unfortunately, the integrals are oscillatory and don't look so nice to
approximate numerically.
Of course, there are various uniqueness theorems that give us
qualitative information about what these nasty solutions must look
like.  Tychonoff's uniqueness theorem says $u(t,x)$ must grow faster
than $e^{cx^2}$ near $x = \infty$, and Widder's theorem says that $u$
must be unbounded above and below for arbitrarily small $t$.  But I'd really like to have an explicit picture to look at.
[1] Rosenbloom, P.C. and Widder, D.V.  A temperature function which
vanishes initially.  American Mathematical Monthly
65(8):607-609, 1958.
 A: I made a MATLAB plot of a partial sum of the first seven terms of the series Carlo referred to. The series is
$$
u(x,t) = \sum_{n=0}^\infty f^{(n)}(t)\frac{x^{2n}}{(2n)!}
$$
where
$$
f(t) = \begin{cases}
\exp(-1/t^2) & t>0\\
 0 & t\leq 0
\end{cases}
$$
The partial sum of the first seven terms looks like this:

It seems like the series is pretty much converged in this domain but I have the feeling that probably some interesting things may happen somewhere else.
One may say that this "wild" solution creeps out of zero so slowly that no one notices... 
A: Here's another plot (using Maple 17, and the first 21 terms) of the solution Carlo referred to.

A: Eq. 1.1 of this 1994 paper gives an explicit example in the form of a series expansion that seems tractable for numerical approximation. At least, I had no difficulty plotting a few terms of the series.
----Update-----
Thanks, Dirk, for the plot. If anyone would like to experiment a bit, you can input this line in Wolfram Alpha for a contour plot
ContourPlot[Sum[D[Exp[-1/t^2],{t,n}]*x^(2*n)/Factorial[2*n],{n,0,7}],{x,0,7},{t,0,7}]
(adjust {n,0,7} {x,0,7} {t,0,7} as desired for more terms in the sum or a different range of x and t)
I notice some curious features, like the oscillations for small t and large x.
