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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.

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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.


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


(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.

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Here's another plot (using Maple 17, and the first 21 terms) of the solution Carlo referred to.

alt text

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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:

alt text

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...

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That should be $x^{2n}$, not $x^2$. – Robert Israel Mar 21 '13 at 18:39
Thanks!$\mbox{}$ – Dirk Mar 21 '13 at 19:17

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