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Maximum principle on the whole space under Gaussian growth

Statement

Assume Countable Choice. Let $T > 0$ , $a \geq 0$ , $A < \infty$ , and let $u \in C ([0, T] \times R^{n}) \cap C^{1, 2} ((0, T] \times R^{n})$ satisfy $u_{t} - Δ u \leq 0, u (t, x) \leq A e^{a ∣ x ∣^{2}} ((t, x) \in [0, T] \times R^{n}) .$ Then $\sup_{[0, T] \times R^{n}} u \leq \sup_{R^{n}} u (0, \cdot) .$ (The same statement holds for a finite union of consecutive strips of length less than $1 / (4 a)$ when $a > 0$ .)

Facts & Assumptions

Given: Countable Choice, $T > 0$ , $a \geq 0$ , $A < \infty$ , and a continuous $u$ on the closed strip, $C^{1, 2}$ in positive time, with $u_{t} - Δ u \leq 0$ and $u (t, x) \leq A e^{a ∣ x ∣^{2}}$ .

[A1]

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

[F1]

The weak maximum principle on a bounded cylinder $B_{R} (0) \times (0, T]$ : a $C^{2, 1}$ subsolution attains its maximum on the parabolic boundary (Weak parabolic maximum principle, Parabolic cylinder and parabolic boundary).

[F2]

The exponential $\exp$ is defined by its series (The real exponential function and the number $e$ by a power series), satisfies $\exp (u + v) = \exp u \exp v$ (The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ ) and $(\exp)^{'} = \exp$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ); the chain, product and quotient rules are those of The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ and Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , and $Δ = \sum_{i} \partial_{i} \partial_{i}$ is the Laplacian of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field.

[F3]

$x^{m} / \exp (a x) \to 0$ as $x \to + \infty$ for every $m$ and $a > 0$ (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ); the Archimedean property supplies the resulting thresholds (Every complete ordered field is Archimedean), and continuous functions on compact sets attain their extrema (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, 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).

Proof

Given: Countable Choice, $T > 0$ , $a \geq 0$ , $A < \infty$ , and $u$ satisfying the subsolution inequality and the Gaussian growth bound.

1.1A1F2F3given

Put $M : = \sup_{R^{n}} u (0, \cdot)$ . If $M = + \infty$ the conclusion is immediate, so assume $M < \infty$ (it is greater than $- \infty$ since the initial trace is real valued). Assume first $a T < 1 / 4$ ; choose $δ > 0$ with $b : = \frac{1}{4 (T + δ)} > a$ , which is possible under this assumption because $1 / (4 T) > a$ and $δ \mapsto 1 / (4 (T + δ))$ is continuous with value $1 / (4 T)$ at $δ = 0$ , and set $B (t, x) : = (T + δ - t)^{- n / 2} e^{∣ x ∣^{2} / [4 (T + δ - t)]}$ for $t \leq T$ ; writing $τ = T + δ - t$ and differentiating, [F2] gives $B_{t} = (\frac{n}{2 τ} + \frac{∣ x ∣^{2}}{4 τ^{2}}) B = Δ B$ , so $(\partial_{t} - Δ) B = 0$ and $v : = u - ε B$ satisfies $(\partial_{t} - Δ) v \leq 0$ for every $ε > 0$ .

2.1step 1.1F2F3given

For every $ε > 0$ there is $R$ with $v (t, x) \leq M$ for all $∣ x ∣ \geq R$ and all $t \in [0, T]$ : indeed $B (t, x) \geq (T + δ)^{- n / 2} e^{b ∣ x ∣^{2}}$ and $u \leq A e^{a ∣ x ∣^{2}}$ , so $v \leq A e^{a ∣ x ∣^{2}} - ε (T + δ)^{- n / 2} e^{b ∣ x ∣^{2}}$ , whose right-hand side tends to $- \infty$ as $∣ x ∣ \to \infty$ because $b > a$ and exponentials dominate constants and polynomials [F3]; hence it is at most the fixed value $M$ for $∣ x ∣ \geq R$ . On the initial slice $v (0, x) = u (0, x) - ε B (0, x) \leq u (0, x) \leq M$ , since $B (0, x) > 0$ .

3.1step 1.1step 2.1F1F3given

Fix $ε > 0$ and $R$ as in step 2.1. For $0 < h < T$ , $v$ has all required derivatives continuous on ${\overline{B}}_{R} \times [h, T]$ , so [F1] applies on this positive-time cylinder. Continuity on ${\overline{B}}_{R} \times [0, T]$ gives $η_{h} : = \max (0, \max_{{\overline{B}}_{R}} v (h, \cdot) - M) \to 0$ as $h ↓ 0$ , because $v (0, \cdot) \leq M$ . Its lateral values are at most $M$ by step 2.1, hence [F1] gives $v \leq M + η_{h}$ for $h \leq t \leq T$ in the ball. At each fixed positive time let $h ↓ 0$ ; combining with step 2.1 outside the ball gives $v \leq M$ on the whole closed strip. Letting $ε ↓ 0$ gives $u \leq M$ when $a T < 1 / 4$ .

4.1step 3.1F3given∎

If $a T \geq 1 / 4$ , choose an integer $N > 4 a T$ and divide $[0, T]$ into the $N$ equal intervals $[k T / N, (k + 1) T / N]$ . Each has positive length $T / N < 1 / (4 a)$ and inherits the same Gaussian bound. Apply the short-strip case to the time-translated function on each interval. On the first interval its supremum is at most $M$ , and inductively the initial supremum of each subsequent strip is at most $M$ . Thus $u \leq M$ throughout $[0, T] \times R^{n}$ , with no time-zero derivative assumption.

Depends on

Used by

Uniqueness for the whole-space heat equation under Gaussian growth Theorem

Dependency tree · two levels

86 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

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)