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Infinite propagation speed for nonnegative heat data

Statement

Assume Countable Choice and let $n \geq 1$ . Let $f \in L^{1} (R^{n})$ satisfy $f \geq 0$ almost everywhere and $f \neq 0$ . Then for every $t > 0$ and every $x \in R^{n}$ the everywhere-defined integral $H_{t} f (x) = \int_{R^{n}} Γ (x - y, t) f (y) d y$ is strictly positive. In particular, if $f$ is compactly supported, nonnegative and nonzero, then the support of the solution at time $t$ is all of $R^{n}$ for every $t > 0$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , $t > 0$ , $x \in R^{n}$ and a representative $f \in L^{1} (R^{n})$ with $f \geq 0$ almost everywhere and $f \neq 0$ .

[A1]

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

[F1]

For every $t > 0$ the heat kernel is strictly positive, $Γ (y, t) > 0$ for all $y$ , and $∥ Γ_{t} ∥_{1} = 1$ (The heat kernel on $R^{n}$ and its causal extension, Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel).

[F2]

For $f \in L^{1} (R^{n})$ , $H_{t} f$ is the $L^{1}$ class of the convolution $Γ_{t} * f$ , and the class is nonnegative almost everywhere when $f \geq 0$ almost everywhere (The heat evolution $H_{t}$ of initial data, Mass conservation and positivity of the heat flow).

[F3]

For a measurable $g \geq 0$ , $\int g = 0$ if and only if $g = 0$ almost everywhere (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

Proof

technique · direct

1.1A1F3given

The set $A : = {y : f (y) > 0}$ is measurable with $λ_{n} (A) > 0$ : were $λ_{n} (A) = 0$ , then $f \leq 0$ almost everywhere together with the hypothesis $f \geq 0$ almost everywhere would make $f = 0$ almost everywhere, contradicting $f \neq 0$ in $L^{1} (R^{n})$ ; equivalently $\int_{R^{n}} f > 0$ by [F3] applied to the nonnegative function $f$ .

2.1step 1.1F1F3given

Fix $t > 0$ and $x \in R^{n}$ . The integrand $y \mapsto Γ (x - y, t) f (y)$ is measurable and nonnegative almost everywhere, by [F1] and the hypothesis on $f$ , and it is strictly positive for every $y \in A$ , since $Γ (x - y, t) > 0$ everywhere by [F1] and $f (y) > 0$ on $A$ ; as $A$ has positive measure by step 1.1, the nonnegative integrand is positive on a set of positive measure, so its integral is strictly positive by [F3].

3.1step 2.1F1F2given

The integral is finite for every $x$ , because $∣ Γ (x - y, t) f (y) ∣ \leq ∥ Γ_{t} ∥_{\infty} ∣ f (y) ∣$ with $∥ f ∥_{1} < \infty$ , so the integral defining $H_{t} f (x)$ converges absolutely at every point and defines the everywhere-positive representative $H_{t} f (x) > 0$ of the class of [F2].

4.1step 3.1given

Consequently, if in addition $supp f$ is compact, then the set where $H_{t} f$ is nonzero is all of $R^{n}$ by step 3.1, so $supp (H_{t} f) = R^{n}$ for every $t > 0$ ; that is, a compactly supported nonnegative nonzero datum has support spreading to the whole space at every positive time.

5.1step 3.1step 4.1given∎

Steps 1.1, 2.1, 3.1 and 4.1 prove strict positivity of the everywhere-defined integral for every nonnegative nonzero $L^{1}$ datum and the full-space support statement for compactly supported data.

Depends on

Used by

The heat equation has no finite propagation speed Counterexample

Dependency tree · two levels

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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)