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The heat equation has no finite propagation speed

Statement refuted

The finite-propagation claim for the heat equation: there is a finite speed $c \geq 0$ such that for every compactly supported datum $g$ and every $t > 0$ the solution $H_{t} g$ vanishes outside the $c t$ -neighbourhood of $supp g$ , that is, $supp (H_{t} g) \subseteq {x : dist (x, supp g) \leq c t}$ . This fails at every positive time for the indicator of an interval.

Facts & Assumptions

Given: Countable Choice, $n = 1$ , the datum $f = 1_{[- 1, 1]}$ , $t > 0$ and $x \in R$ .

[A1]

Countable Choice is the hypothesis carried by the evolution and positivity suppliers below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The indicator of a measurable set is measurable (An indicator function is measurable exactly when its set is measurable), and $supp (f) = [- 1, 1]$ for $f = 1_{[- 1, 1]}$ .

[F2]

If $g \in L^{1} (R)$ satisfies $g \geq 0$ almost everywhere and $g \neq 0$ , then for every $t > 0$ the everywhere-defined integral $H_{t} g (x) = \int_{R} Γ (x - y, t) g (y) d y$ is strictly positive at every $x \in R$ (Infinite propagation speed for nonnegative heat data).

[F3]

For $t > 0$ the heat kernel is $Γ (z, t) = (4 π t)^{- 1 / 2} e^{- z^{2} / (4 t)} > 0$ with unit mass, and $H_{t}$ is the evolution of The heat evolution $H_{t}$ of initial data (The heat kernel on $R^{n}$ and its causal extension).

Counterexample

technique · direct

1.1A1F1givenalgebra

The datum $f = 1_{[- 1, 1]}$ is measurable by [F1], is nonnegative everywhere with $f = 1$ on a set of measure $2$ , is not the zero class, has $\int_{R} ∣ f ∣ = 2 < \infty$ , hence lies in $L^{1} (R)$ , and has compact support $[- 1, 1]$ .

2.1F2F3step 1.1given

Infinite propagation: since $f \in L^{1} (R)$ , $f \geq 0$ and $f \neq 0$ , the infinite-propagation corollary [F2] gives $H_{t} f (x) > 0$ at every $x \in R$ and every $t > 0$ ; consequently the set on which the solution is nonzero is all of $R$ , so $supp (H_{t} f) = R$ for every $t > 0$ .

3.1step 2.1givenalgebra

Fix any finite speed $c \geq 0$ and any $t > 0$ . At $x = 2 + c t$ one has $dist (x, [- 1, 1]) = 1 + c t > c t$ , but $H_{t} f (x) > 0$ by step 2.1. Thus the required support inclusion fails for every proposed finite speed.

4.1step 2.1step 3.1given∎

Steps 1.1, 2.1 and 3.1 exhibit a nonzero nonnegative compactly supported $L^{1}$ datum whose heat flow is strictly positive at every point at every positive time, so the support of the solution is the whole line although the support of the datum is $[- 1, 1]$ ; the finite-propagation claim is refuted.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #5: The Fundamental Solution for the Heat Equation (Fall 2011) (standard reference, not scraped)