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The final-time face is not part of the parabolic boundary

Statement refuted

The claim refuted is that every supersolution $u_{t} - Δ u \geq 0$ on a bounded cylinder $Q$ satisfies $\max_{\overline{Q}} u \leq \max_{\partial_{p} Q} u$ . The witness has its larger maximum at a spatially interior point of the final-time face, which is excluded from the parabolic boundary. Take $Ω = (0, π)$ and $u (x, t) : = e^{t} \sin x .$ Then $u_{t} = e^{t} \sin x$ and $Δ u = - e^{t} \sin x$ , so $u_{t} - Δ u = 2 e^{t} \sin x \geq 0$ in $Q$ : $u$ is a supersolution. Its maximum over the closed cylinder is $\max_{\overline{Q}} u = e^{T}$ , attained at the interior point $(π / 2, T)$ of the final-time face, whereas $\max_{\partial_{p} Q} u = \max (\max_{[0, π]} \sin, 0) = 1 < e^{T}$ because the lateral data vanish and the initial data are $\sin x \leq 1$ with equality at $x = π / 2$ . So the final-time face is not part of the parabolic boundary, and for a supersolution the maximum over the cylinder is genuinely larger than the parabolic-boundary maximum: the maximum principle is sign-sensitive and does not extend in the reverse direction.

Facts & Assumptions

Given: $T > 0$ , the cylinder $Q = (0, π) \times (0, T]$ with $\overline{Q} = [0, π] \times [0, T]$ , and the function $u (x, t) = e^{t} \sin x$ .

[F1]

The cylinder vocabulary: $\partial_{p} Q = (\overline{Ω} \times {0}) \cup (\partial Ω \times [0, T])$ , no point of the final-time face belongs to $\partial_{p} Q$ , and $u_{t} - Δ u \geq 0$ in $Q$ is imposed for $0 < t \leq T$ , with $u_{t}$ interpreted as the left time derivative at $t = T$ (Parabolic cylinder and parabolic boundary).

[F2]

$\sin$ and $\cos$ are $C^{\infty}$ with $(\sin x)^{'} = \cos x$ and $(\cos x)^{'} = - \sin x$ , $\sin 0 = 0$ , $\cos 0 = 1$ (The derivatives of sine and cosine are cosine and minus sine, Sine and cosine defined by their real power series); $\sin (π / 2) = 1$ , $\sin π = 0$ , and $\sin x > 0$ for $0 < x < π$ (Quarter-turn values and shifts by pi/2 and pi, Pi is the first positive zero of sine), while $\sin$ has range $[- 1, 1]$ (Signs, monotonicity intervals, and ranges of sine and cosine).

[F3]

$\exp$ is $C^{\infty}$ with $\exp^{'} = \exp$ and $\exp (x) > 0$ for every real $x$ (The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ); the mean value theorem applies to $\exp$ on $[0, T]$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F4]

The Laplacian on $R^{1}$ is $Δ f = f^{''}$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field), and the weak maximum principle on a bounded cylinder: a subsolution $v \in C^{2, 1} (\overline{Q})$ with $v_{t} - Δ v \leq 0$ in $Q$ satisfies $\max_{\overline{Q}} v = \max_{\partial_{p} Q} v$ (Weak parabolic maximum principle).

Counterexample

Given: $T > 0$ , the cylinder $Q = (0, π) \times (0, T]$ , and $u (x, t) = e^{t} \sin x$ .

1.1F2F3F4given

The function $u$ is smooth on $\overline{Q}$ (a product of the smooth functions $e^{t}$ and $\sin x$ , [F2] and [F3]), with $u_{t} = e^{t} \sin x$ and $Δ u = u_{x x} = - e^{t} \sin x$ by [F2] and [F4]; hence $u_{t} - Δ u = 2 e^{t} \sin x \geq 0$ on $Q$ , because $e^{t} > 0$ by [F3] and $\sin x > 0$ for $0 < x < π$ by [F2].

1.2F1F2given

On the parabolic boundary, $u (0, t) = u (π, t) = e^{t} \sin π = 0$ for every $t \in [0, T]$ , and $u (x, 0) = \sin x \leq 1$ with $u (π / 2, 0) = 1$ ; hence $\max_{\partial_{p} Q} u = 1$ .

1.3F2F3given

For every $(x, t) \in \overline{Q}$ one has $\sin x \leq 1$ by [F2] and $e^{t} \leq e^{T}$ : indeed $e^{T} - e^{t} = e^{c} (T - t) > 0$ for some $c \in (t, T)$ whenever $t < T$ by [F3] and the mean value theorem, so $u (x, t) \leq e^{T}$ ; equality holds exactly at $(x, t) = (π / 2, T)$ , where $u = e^{T} \sin (π / 2) = e^{T}$ . Thus $\max_{\overline{Q}} u = e^{T}$ , attained at the point $(π / 2, T)$ of the final-time face.

2.1step 1.2step 1.3F1F3given

The point $(π / 2, T)$ does not belong to $\partial_{p} Q$ , because $π / 2 \in (0, π)$ and $T > 0$ by [F1]; by step 1.3 it is a point of $\overline{Q}$ at which $u$ attains the value $e^{T}$ , while by step 1.2 the parabolic boundary carries the strictly smaller maximum $1$ . Since $e^{T} > 1$ (again by the mean value theorem applied to $\exp$ on $[0, T]$ , [F3]), a supersolution has its maximum over $\overline{Q}$ strictly larger than the parabolic-boundary maximum, refuting the proposed supersolution maximum bound. The minimum principle for supersolutions, obtained by applying the weak maximum principle to $- u$ , remains valid.

3.1step 1.1step 1.2F1F4given∎

The sign sensitivity is real and the weak maximum principle is not contradicted: $v : = - u$ satisfies $v_{t} - Δ v = - 2 e^{t} \sin x \leq 0$ on $Q$ , and by [F4] its maximum over $\overline{Q}$ equals $\max_{\partial_{p} Q} v = 0$ , attained on the lateral faces, consistently with the theorem being stated for subsolutions.

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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)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext, Springer 2011) (standard reference, not scraped)