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Supremum norm stability for forced heat problems

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be nonempty, bounded and open, $T > 0$ , and let real $u, v \in C^{2, 1} (\overline{Ω} \times [0, T])$ solve $u_{t} - Δ u = f$ , $v_{t} - Δ v = g$ with $f, g \in C (\overline{Ω} \times [0, T])$ . Put $a = ∥ u (\cdot, 0) - v (\cdot, 0) ∥_{\infty}$ , $b_{t} = \sup_{\partial Ω \times [0, t]} ∣ u - v ∣$ , and $m (s) = \max_{\overline{Ω}} ∣ f (\cdot, s) - g (\cdot, s) ∣$ . Then for $0 < t \leq T$ , $\sup_{\overline{Ω} \times [0, t]} ∣ u - v ∣ \leq \max (a, b_{t}) + \int_{0}^{t} m (s) d s,$ and hence also the bound with $a + b_{t}$ in place of the maximum.

Facts & Assumptions

Given: Countable Choice, a bounded open $Ω \subseteq R^{n}$ , $T > 0$ , real $u, v \in C^{2, 1} (\overline{Q})$ on $Q = Ω \times (0, T]$ with $u_{t} - Δ u = f$ and $v_{t} - Δ v = g$ in $Q$ for continuous $f, g$ , and $0 < t \leq T$ .

[A1]

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

[F1]

Cylinder convention and parabolic boundary: the class $C^{2, 1} (\overline{Q})$ , the closed cylinder $\overline{Ω} \times [0, T]$ and $\partial_{p} (Ω \times (0, t]) = (\overline{Ω} \times {0}) \cup (\partial Ω \times [0, t])$ are those of Parabolic cylinder and parabolic boundary.

[F2]

Comparison: if $U, V \in C^{2, 1} (\overline{Q})$ with $Q = Ω \times (0, t]$ , $U_{t} - Δ U \leq V_{t} - Δ V$ in $Q$ and $U \leq V$ on $\partial_{p} Q$ , then $U \leq V$ on $\overline{Q}$ (Comparison and uniqueness for the bounded-cylinder heat problem).

[F3]

$\overline{Q}$ is compact and continuous real functions on nonempty compact sets attain their extrema (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Open cover, subcover, compact metric space, and compact subset of a metric space, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value); continuous maps on compact metric spaces are uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, Uniform continuity of a map of metric spaces: one $δ$ serving every point).

[F4]

Fundamental theorem of calculus, first part: if $φ$ is continuous on $[0, t]$ , then $s \mapsto \int_{0}^{s} φ$ is differentiable with derivative $φ$ (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive), continuity on a compact interval supplying the integrability used to form the integral (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

[F5]

The Laplacian is $Δ w = \sum_{i} \partial_{i} \partial_{i} w$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field), the partial derivatives being those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ; a function of the time variable alone has all its $x$ -partial derivatives $0$ , and $\pm m$ is an admissible forcing for the comparison principle.

Proof

Given: Countable Choice, bounded $Ω$ , $T > 0$ , solutions $u, v \in C^{2, 1} (\overline{Q})$ of $u_{t} - Δ u = f$ , $v_{t} - Δ v = g$ , continuous $f, g$ , and $0 < t \leq T$ .

1.1A1F1F3given

Put $w : = u - v$ and $Q_{t} : = Ω \times (0, t]$ . Then $w \in C^{2, 1} (\overline{Q_{t}})$ with $w_{t} - Δ w = f - g$ in $Q_{t}$ , and on the parabolic boundary of $Q_{t}$ one has $∣ w ∣ \leq a$ on $\overline{Ω} \times {0}$ and $∣ w ∣ \leq b_{t}$ on $\partial Ω \times [0, t]$ by the definitions of $a$ and $b_{t}$ ; moreover the function $s \mapsto m (s) = \max_{\overline{Ω}} ∣ f (\cdot, s) - g (\cdot, s) ∣$ is well defined on $[0, t]$ and continuous there: it is a maximum of a continuous function on the compact $\overline{Ω}$ for each $s$ by [F3], and given $ε > 0$ , uniform continuity of $f$ and $g$ on the compact $\overline{Q}$ [F3] gives $δ > 0$ such that $∣ f (x, s) - f (x, s_{0}) ∣ < ε / 2$ and $∣ g (x, s) - g (x, s_{0}) ∣ < ε / 2$ for all $x \in \overline{Ω}$ whenever $∣ s - s_{0} ∣ < δ$ , whence $∣ m (s) - m (s_{0}) ∣ \leq ε$ .

2.1step 1.1F4F5given

Let $M : = \max (a, b_{t}) \geq 0$ and for $s \in [0, t]$ put $W_{+} (s) : = M + \int_{0}^{s} m (σ) d σ$ and $W_{-} (s) : = - M - \int_{0}^{s} m (σ) d σ$ , viewed as functions of $(s, x)$ . By [F4] and the continuity of $m$ from step 1.1, both are of class $C^{2, 1} (\overline{Q_{t}})$ with $(W_{\pm})_{t} - Δ W_{\pm} = \pm m$ : the time derivative is $\pm m (s)$ by [F4] and every $x$ -partial derivative vanishes because $W_{\pm}$ depends on $s$ alone, so $Δ W_{\pm} = 0$ by [F5]. On the parabolic boundary of $Q_{t}$ one has $W_{-} \leq - M \leq w \leq M \leq W_{+}$ , since $∣ w ∣ \leq M$ there by step 1.1.

3.1step 1.1step 2.1F2given

Comparison applied twice. For the pair $(U, V) = (w, W_{+})$ : $w_{t} - Δ w - (W_{+, t} - Δ W_{+}) = (f - g) - m \leq 0$ in $Q_{t}$ and $w \leq W_{+}$ on $\partial_{p} Q_{t}$ by step 2.1, so [F2] gives $w \leq W_{+}$ on $\overline{Q_{t}}$ . For the pair $(U, V) = (W_{-}, w)$ : $W_{-, t} - Δ W_{-} - (w_{t} - Δ w) = - m - (f - g) \leq 0$ in $Q_{t}$ and $W_{-} \leq w$ on $\partial_{p} Q_{t}$ by step 2.1, so [F2] gives $W_{-} \leq w$ on $\overline{Q_{t}}$ . Therefore $∣ w (s, x) ∣ \leq M + \int_{0}^{s} m (σ) d σ$ for every $(s, x) \in [0, t] \times \overline{Ω}$ .

4.1step 3.1given∎

Since $s \mapsto \int_{0}^{s} m$ is nondecreasing on $[0, t]$ (the integrand is nonnegative), step 3.1 gives $∣ u - v ∣ \leq M + \int_{0}^{t} m$ on $\overline{Ω} \times [0, t]$ , hence $\sup_{\overline{Ω} \times [0, t]} ∣ u - v ∣ \leq \max (a, b_{t}) + \int_{0}^{t} m (s) d s$ . Finally $\max (a, b_{t}) \leq a + b_{t}$ , so the same estimate holds with $a + b_{t}$ in place of the maximum.

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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)