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Strict positivity propagates to later interior times

Statement

Assume Countable Choice. Let $Ω$ be bounded and connected and $u \in C^{2, 1} (\overline{Ω} \times [0, T])$ be a nonnegative homogeneous heat solution. If $u (x_{*}, s_{*}) > 0$ at an interior point with $0 < s_{*} < T$ , then $u (x, t) > 0$ for every $x \in Ω$ and $s_{*} < t \leq T$ . The same conclusion for every $0 < t \leq T$ holds if the continuous initial trace is positive at some interior point. Nontriviality only at a later time does not assert positivity before that time.

Facts & Assumptions

Given: Countable Choice, a bounded connected open $Ω \subseteq R^{n}$ , $T > 0$ , and a nonnegative $u \in C^{2, 1} (\overline{Q})$ on $Q = Ω \times (0, T]$ with $u_{t} - Δ u = 0$ in $Q$ .

[A1]

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

[F1]

Strong parabolic maximum principle: on a bounded parabolic cylinder, a subsolution attaining its maximum $M$ at an interior point $(x_{0}, t_{0})$ with $x_{0} \in Ω$ and $0 < t_{0} \leq T$ is equal to $M$ on $Ω_{0} \times (0, t_{0}]$ , where $Ω_{0}$ is the connected component of $Ω$ containing $x_{0}$ (Strong parabolic maximum principle).

[F2]

The cylinder $Q = Ω \times (0, T]$ and the class $C^{2, 1} (\overline{Q})$ are those of Parabolic cylinder and parabolic boundary; in particular $u$ is continuous on $\overline{Q}$ and $u_{t} - Δ u = 0$ on the open cylinder.

[F3]

Connectedness and components: $C (x)$ is the largest connected subset of $X$ containing $x$ (Connected components, quasicomponents, and totally disconnected spaces), so a connected space $X$ has $C (x) = X$ for every $x \in X$ since $X$ itself is then a connected subset containing $x$ (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets); in particular the component $Ω_{0}$ of [F1] equals $Ω$ for every $x_{0} \in Ω$ .

[F4]

Continuity at a point: for real $ε > 0$ there is $δ > 0$ with $∣ u (x, s) - u (x_{0}, 0) ∣ < ε$ whenever $(x, s) \in \overline{Q}$ is within $δ$ of $(x_{0}, 0)$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

Proof

Given: Countable Choice, a bounded connected $Ω$ , and a nonnegative solution $u \in C^{2, 1} (\overline{Q})$ with $u_{t} - Δ u = 0$ in $Q$ .

1.1A1F1F2F3given

Let $t_{1} \in (s_{*}, T]$ and $x_{1} \in Ω$ with $u (x_{1}, t_{1}) = 0$ . On the truncated cylinder $Q_{1} : = Ω \times (0, t_{1}]$ put $v : = - u$ ; then $v \in C^{2, 1} (\overline{Q_{1}})$ , $v_{t} - Δ v = 0$ in $Q_{1}$ , and $v \leq 0$ on $\overline{Q_{1}}$ because $u \geq 0$ , so the maximum $M = 0$ of $v$ over $\overline{Q_{1}}$ is attained at $(x_{1}, t_{1})$ with $x_{1} \in Ω$ and $0 < t_{1} \leq t_{1}$ ; since $Ω$ is connected, [F3] makes the relevant component all of $Ω$ , and [F1] gives $v = 0$ on $Ω \times (0, t_{1}]$ , that is $u = 0$ there. But $s_{*} < t_{1}$ and $x_{*} \in Ω$ , so $u (x_{*}, s_{*}) = 0$ , contradicting the hypothesis $u (x_{*}, s_{*}) > 0$ . Hence $u (x, t) > 0$ for every $x \in Ω$ and $s_{*} < t \leq T$ .

2.1step 1.1F2F4given

Suppose now that the continuous initial trace satisfies $u (x_{0}, 0) > 0$ at some interior point $x_{0} \in Ω$ , and let $t \in (0, T]$ be given. Choose $s_{*} \in (0, \min (t, T))$ ; by [F4] applied with $ε = u (x_{0}, 0) / 2 > 0$ there is $δ > 0$ with $u (x, s) > u (x_{0}, 0) / 2 > 0$ for every $(x, s) \in \overline{Q}$ with $∣ (x, s) - (x_{0}, 0) ∣ < δ$ , and the point $(x_{0}, s_{*})$ qualifies for $s_{*}$ small enough, so it is admissible in step 1.1; since $s_{*} < t$ , step 1.1 gives $u (x, t) > 0$ for every $x \in Ω$ . As $t \in (0, T]$ was arbitrary, a positive interior point of the initial trace forces strict positivity at all later interior space-time points.

3.1step 1.1step 2.1given∎

Steps 1.1 and 2.1 establish both positivity clauses of the statement, together with the recorded caveat: positivity of $u$ at one interior time $s_{*}$ propagates only to later times $t > s_{*}$ , and no claim is made about times before $s_{*}$ ; no global lower bound, boundary positivity or uniqueness statement is asserted. The argument uses no choice beyond the Countable Choice declared in [A1].

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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)
Sung-Jin Oh, Lecture Notes for Math 222A (19 March 2024) (standard reference, not scraped)