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The backward heat solution map is unbounded

Statement

Assume Countable Choice. Let $T > 0$ and consider the heat equation $u_{t} = u_{x x}$ on $(0, π) \times (0, T]$ with homogeneous Dirichlet boundary data $u (0, t) = u (π, t) = 0$ .

Well-definedness of the terminal-to-initial map. If a $C^{4, 2}$ function $w$ on $[0, π] \times [0, T]$ solves this problem with $w (x, T) = 0$ for all $x$ , then $w \equiv 0$ (Backward uniqueness for the heat equation on a bounded interval). Hence on the class of $C^{4, 2}$ solutions the terminal data determine the solution, and the map $S$ sending a terminal datum $u (\cdot, T)$ to the initial state $u (\cdot, 0)$ is well defined.

Unboundedness. For every integer $k \geq 1$ the function $u_{k} (x, t) : = e^{k^{2} (T - t)} \sin (k x)$ is a classical solution of $u_{t} = u_{x x}$ on $(0, π) \times (0, T]$ with $u_{k} (0, t) = u_{k} (π, t) = 0$ , initial state $u_{k} (\cdot, 0) = e^{k^{2} T} \sin (k \cdot)$ and terminal data $u_{k} (\cdot, T) = \sin (k \cdot)$ . Consequently there is no constant $C < \infty$ with $∥ u (\cdot, 0) ∥_{L^{2} (0, π)} \leq C ∥ u (\cdot, T) ∥_{L^{2} (0, π)}$ for all $C^{4, 2}$ solutions $u$ of this problem: the solutions ${\tilde{u}}_{k} (x, t) = e^{- k^{2} t} \sin (k x)$ have terminal data $g_{k} = e^{- k^{2} T} \sin (k \cdot)$ with $∥ g_{k} ∥_{2} = \sqrt{π / 2} e^{- k^{2} T} \to 0$ while $∥ {\tilde{u}}_{k} (\cdot, 0) ∥_{2} = \sqrt{π / 2}$ for every $k$ . Since $S g_{k} = \sin (k \cdot) = e^{k^{2} T} g_{k}$ , the terminal-to-initial map multiplies the $k$ -th sine mode by $e^{k^{2} T}$ and is unbounded with respect to the $L^{2} (0, π)$ norms on its domain and range: the backward heat problem on a bounded interval has no norm-stable solution operator, and the amplification factor of the $k$ -th Dirichlet mode is exactly $e^{k^{2} T}$ .

Facts & Assumptions

Given: Countable Choice, $T > 0$ , the interval $(0, π)$ , and an integer $k \geq 1$ .

[A1]

Countable Choice is the ambient hypothesis, inherited through the backward-uniqueness and $L^{2}$ suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Backward uniqueness: a $C^{4, 2}$ function on $[0, π] \times [0, T]$ satisfying $w_{t} = w_{x x}$ , $w (0, t) = w (π, t) = 0$ and $w (x, T) = 0$ vanishes identically (Backward uniqueness for the heat equation on a bounded interval).

[F2]

On the interval the heat operator is $\partial_{t} - \partial_{x}^{2}$ (The heat operator, the heat equation, and the Cauchy problem, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field); the exponential has derivative itself (The exponential function is smooth and $(\exp)^{'} = \exp$ ) and compositions use The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ; sine and cosine satisfy $(\sin)^{'} = \cos$ , $(\cos)^{'} = - \sin$ (The derivatives of sine and cosine are cosine and minus sine) and $\sin (m π) = 0$ for every integer $m$ (The zero sets of sine and cosine and the least positive common period 2 pi).

[F3]

The $L^{2} (0, π)$ inner product is $⟨ f, g ⟩ = \int_{0}^{π} f g$ on the quotient space of The space $L^{p} (μ)$ as the quotient by null functions, with $∥ f ∥_{2}^{2} = \int_{0}^{π} f^{2}$ ( $L^{2}$ with the integral pairing is a Hilbert space), and $∥ \sin (k \cdot) ∥_{2} = \sqrt{π / 2}$ (L2 normalisation of the sine modes on an interval).

[F4]

$e^{- k^{2} T} \to 0$ as $k \to \infty$ , faster than every polynomial (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ).

Proof

Given: Countable Choice, $T > 0$ and $k \geq 1$ .

1.1A1F1given

If $u_{1}, u_{2}$ are $C^{4, 2}$ solutions with $u_{i} (0, t) = u_{i} (π, t) = 0$ and the same terminal data $u_{1} (\cdot, T) = u_{2} (\cdot, T)$ , then $w : = u_{1} - u_{2}$ is $C^{4, 2}$ and satisfies $w_{t} = w_{x x}$ , $w (0, t) = w (π, t) = 0$ and $w (x, T) = 0$ , so [F1] gives $w \equiv 0$ ; hence the terminal-to-initial map $S$ is well defined on the terminal data of the class.

1.2F2given

For every $k \geq 1$ the function $u_{k} (x, t) = e^{k^{2} (T - t)} \sin (k x)$ has $\partial_{t} u_{k} = - k^{2} u_{k}$ and, by [F2], $\partial_{x}^{2} u_{k} = - k^{2} \sin (k x) e^{k^{2} (T - t)} = - k^{2} u_{k}$ , with boundary values $u_{k} (0, t) = \sin 0 = 0$ and $u_{k} (π, t) = \sin (k π) = 0$ ; the same computation with $e^{- k^{2} t}$ shows that ${\tilde{u}}_{k} (x, t) = e^{- k^{2} t} \sin (k x)$ is a solution with ${\tilde{u}}_{k} (\cdot, 0) = \sin (k \cdot)$ and terminal data $g_{k} = e^{- k^{2} T} \sin (k \cdot)$ .

2.1step 1.2F3given

By [F3] and scaling by the positive factor $e^{- k^{2} T}$ , $∥ g_{k} ∥_{2} = e^{- k^{2} T} ∥ \sin (k \cdot) ∥_{2} = \sqrt{π / 2} e^{- k^{2} T}$ , while $∥ {\tilde{u}}_{k} (\cdot, 0) ∥_{2} = ∥ \sin (k \cdot) ∥_{2} = \sqrt{π / 2}$ .

2.2step 1.2F4given

By [F4] the sequence $e^{- k^{2} T}$ tends to $0$ ; consequently the terminal data $g_{k}$ of step 1.2 satisfy $∥ g_{k} ∥_{2} = \sqrt{π / 2} e^{- k^{2} T} \to 0$ while $∥ {\tilde{u}}_{k} (\cdot, 0) ∥_{2} = \sqrt{π / 2}$ stays constant.

3.1step 1.1step 1.2step 2.2F3given∎

Since $g_{k}$ are admissible terminal data with $∥ g_{k} ∥_{2} \to 0$ but $S g_{k} = \sin (k \cdot)$ and $∥ \sin (k \cdot) ∥_{2} = \sqrt{π / 2}$ for every $k$ , no finite constant $C$ satisfies $∥ u (\cdot, 0) ∥_{2} \leq C ∥ u (\cdot, T) ∥_{2}$ on the class, so $S$ is unbounded; explicitly $S$ multiplies the $k$ -th sine mode by $e^{k^{2} T}$ , which is the amplification factor asserted.

Depends on

Used by

Backward heat amplifies small high-frequency errors Counterexample
Backward ill-posedness does not mean universal nonexistence Remark

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Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)