TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Uniqueness for the whole-space heat equation under Gaussian growth

Statement

Assume Countable Choice. Let $T > 0$ , $C < \infty$ , $a \geq 0$ , and let $u \in C ([0, T] \times R^{n}) \cap C^{1, 2} ((0, T] \times R^{n})$ satisfy $u_{t} - Δ u = 0$ on $R^{n} \times (0, T)$ , $u (0, \cdot) = 0$ and $∣ u (t, x) ∣ \leq C e^{a ∣ x ∣^{2}} ((t, x) \in [0, T] \times R^{n}) .$ Then $u \equiv 0$ .

Facts & Assumptions

Given: Countable Choice, $T > 0$ , $C < \infty$ , $a \geq 0$ , and $u$ in the stated class with $u_{t} - Δ u = 0$ on $R^{n} \times (0, T)$ , $u (0, \cdot) = 0$ and $∣ u ∣ \leq C e^{a ∣ x ∣^{2}}$ on the closed strip.

[A1]

Countable Choice is the ambient hypothesis (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Whole-space maximum principle under Gaussian growth: if $w \in C ([0, T] \times R^{n}) \cap C^{1, 2} ((0, T] \times R^{n})$ satisfies $w_{t} - Δ w \leq 0$ and $w (t, x) \leq A e^{a ∣ x ∣^{2}}$ on the closed strip, then $\sup_{[0, T] \times R^{n}} w \leq \sup_{R^{n}} w (0, \cdot)$ , the finite subdivision into strips being available by the Archimedean property (Maximum principle on the whole space under Gaussian growth, Every complete ordered field is Archimedean).

[F2]

The heat operator $\partial_{t} - Δ$ is linear: for a constant $α$ and a $C^{1, 2}$ function $w$ , $(α w)_{t} = α w_{t}$ and $Δ (α w) = α Δ w$ , by the constant-multiple rule applied to the one-variable derivatives along lines and the sum formula $Δ = \sum_{i} \partial_{i} \partial_{i}$ (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

Proof

Given: Countable Choice, $T > 0$ , $C < \infty$ , $a \geq 0$ , and $u$ satisfying $u_{t} - Δ u = 0$ on the open strip, $u (0, \cdot) = 0$ and $∣ u ∣ \leq C e^{a ∣ x ∣^{2}}$ on the closed strip.

1.1A1F1given

The function $u$ satisfies the hypotheses of [F1]: it lies in the stated class, $u_{t} - Δ u = 0 \leq 0$ on $R^{n} \times (0, T)$ , and $u \leq ∣ u ∣ \leq C e^{a ∣ x ∣^{2}}$ on the closed strip. Hence [F1] gives $\sup_{[0, T] \times R^{n}} u \leq \sup_{R^{n}} u (0, \cdot) = 0$ , that is $u \leq 0$ everywhere on the strip.

2.1step 1.1F1F2given

By [F2] the function $- u$ also lies in the class, with $(- u)_{t} - Δ (- u) = - (u_{t} - Δ u) = 0 \leq 0$ on the open strip and $(- u) (0, \cdot) = 0$ ; moreover $- u \leq ∣ u ∣ \leq C e^{a ∣ x ∣^{2}}$ on the closed strip. So [F2] and [F1] apply to $- u$ and give $\sup (- u) \leq \sup (- u) (0, \cdot) = 0$ , that is $u \geq 0$ everywhere on the strip.

3.1step 1.1step 2.1F1given∎

Steps 1.1 and 2.1 give $u \leq 0 \leq u$ on $[0, T] \times R^{n}$ , hence $u \equiv 0$ ; the only choices made are the finite subdivisions supplied by the Archimedean property inside [F1], and Countable Choice is inherited from that supplier.

Depends on

Used by

Duhamel principle for the whole-space heat equation Theorem
The inhomogeneous heat Cauchy formula Theorem

Dependency tree · two levels

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)