ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedaudited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Semilinear characteristics with logistic growth

Example

Let $g : R \to R$ be $C^{1}$ . For $u_{t} + u_{x} = u (1 - u)$ and $u (0, x) = g (x)$ , the characteristic formula is

$u (t, x) = \frac{g (x - t) e^{t}}{1 + g (x - t) (e^{t} - 1)},$

on the open set where $1 + g (x - t) (e^{t} - 1) > 0$ . For each fixed label $ξ = x - t$ , this is the time interval containing zero before any pole.

Facts & Assumptions

Given: A $C^{1}$ datum $g : R \to R$ and a point $(t, x)$ where the displayed denominator is positive.

Verification

technique · direct

1.1givenalgebra

The projected characteristics in the independent-variable plane are $X (t, ξ) = (t, ξ + t)$ . Thus their spatial component is $ξ + t$ , so $ξ = x - t$ , and the Jacobian of $(t, ξ) \mapsto (t, ξ + t)$ is $1$ .

2.1step 1.1algebra

Along one such curve, $z^{'} = z (1 - z)$ and $z (0) = g (ξ)$ ; put $q = g (ξ)$ and $D (t) = 1 + q (e^{t} - 1)$ . On $D (t) > 0$ , direct differentiation gives $z = q e^{t} / D$ and $z^{'} = q e^{t} (1 - q) / D^{2} = z (1 - z)$ , with $z (0) = q$ . This includes $q = 0$ and $q = 1$ without division by either. Since $D (0) = 1$ and $D^{'} (t) = q e^{t}$ has constant sign, ${t : D (t) > 0}$ is exactly the interval containing zero on which the denominator is nonzero.

3.1step 2.1given∎

Substitute $ξ = x - t$ to obtain the formula, and characteristic reconstruction verifies the PDE on its stated domain.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

First order PDE: The Methods of Characteristics (standard reference, not scraped)