---
tag: RN4J24CD83
kind: counterexample
name: Tychonoff's non-uniqueness example
msc: [35K05, 35A02, 35K15]
uses: [VP2HN86VAG]
sources:
  - url: https://en.wikipedia.org/wiki/Heat_equation
---

## Counterexample

There is a function $u \colon \R \times \R \to \R$, smooth and not identically zero, such that
$$\partial_t u = \partial_x^2 u \ \text{on } \R \times \R \quad \text{and} \quad u(x, t) = 0 \ \text{for all } t \le 0.$$
So the Cauchy problem for the [[heat equation|VP2HN86VAG]] on $\R \times (0, \infty)$ with initial value $0$ has a solution other than $u = 0$.
One construction: fix $\alpha > 1$, let $g(t) = \exp(-t^{-\alpha})$ for $t > 0$ and $g(t) = 0$ for $t \le 0$, and set
$$u(x, t) = \sum_{k=0}^{\infty} g^{(k)}(t) \, \frac{x^{2k}}{(2k)!}.$$
Term by term, $\partial_x^2$ of the $k$-th term is the $(k-1)$-th term with $g^{(k-1)}$ replaced by $g^{(k)}$, so formally $\partial_x^2 u = \sum_k g^{(k+1)}(t) x^{2k}/(2k)! = \partial_t u$.
There are constants $\theta, c > 0$ with $\abs{g^{(k)}(t)} \le k! \, (\theta t)^{-k} \exp(-c \, t^{-\alpha})$ for $t > 0$.
These bounds show that the series and its differentiated series converge locally uniformly, so $u$ is a genuine solution.
Since $g$ is not identically zero, neither is $u$.

## Remarks

The solution grows very fast in $x$: on no strip $\R \times (0, T)$ is it bounded by $A e^{a x^2}$ for constants $A$ and $a$.
Tychonoff showed that uniqueness does hold in the class of solutions with $\abs{u(x, t)} \le A e^{a \abs{x}^2}$.
So a growth condition at infinity is needed for uniqueness on the whole line; on a bounded interval with boundary values, the [[maximum principle|EMDVD4E83K]] gives uniqueness without one.
Nonnegative solutions are also unique, by Widder's theorem, so $u$ must change sign.

## Explanations

### L1 human

One would expect a rod that starts at temperature zero everywhere and receives no heat to stay at zero.
On an infinite rod this is false in a strange way: heat can arrive from infinity, through temperatures that grow enormously far away.
The example shows that the heat equation alone does not determine the future; one must also rule out such wild behaviour at infinity.