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.