Strong parabolic maximum principle

Given: Countable Choice, a parabolic cylinder $Q=\mathrm{\Omega }\times (0,T]$ with $\mathrm{\Omega }$ bounded, $u\in C^{2,1}(\bar{Q})$ with $u_t-\mathrm{\Delta }u\le 0$ in $Q$, and a point $(x_0,t_0)$ with $x_0\in \mathrm{\Omega }$, $0<t_0\le T$ and $u(x_0,t_0)=M:={\max }_{\bar{Q}}u$.

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

Submean inequality: if $u$ is $C^{2,1}$ on a neighbourhood of a closed heat ball $E_{\eta }(P)$ and $u_t-\mathrm{\Delta }u\le 0$ there, then $u(P)\le \frac{1}{2{\eta }^n}{\int }_{t_P-{\eta }^2\mathrm{/}4\pi }^{t_P}\frac{{\rho }_n(t_P-s)}{t_P-s}{\int }_{\mathrm{\mid }y-x_P\mathrm{\mid }={\rho }_n(t_P-s)}u(y,s)dS(y)ds,$ and $u(P)\le M$ whenever $u\le M$ on $E_{\eta }(P)$ (Submean inequality for heat subsolutions on heat balls).

Time slices of heat balls: for $0<\tau <r^2\mathrm{/}(4\pi )$ the time slice of $E_r(t,x)$ is the closed ball of radius ${\rho }_n(\tau )=\sqrt{2n\tau \log \frac{r^2}{4\pi \tau }}$; the slice at $\tau =r^2\mathrm{/}(4\pi )$ is the single point $\{x\}$ and the slice is empty for $\tau >r^2\mathrm{/}(4\pi )$, so $E_r(t,x)\subseteq Q$ forces $t-r^2\mathrm{/}(4\pi )>0$; the spatial projection of $E_r(t,x)$ then lies compactly inside the open set $\mathrm{\Omega }$ (Heat balls and their time slices).

The level set $L_r(P):=\{(y,s):s<t_P,\mathrm{\Gamma }(x_P-y,t_P-s)=r^{-n}\}$ is the lateral part of $\mathrm{\partial }{E}_r(P)$, it is a $C^{\mathrm{\infty }}$ hypersurface on which $\mathrm{\nabla }\mathrm{\Gamma }\mathrm{\ne }0$, and the top point $P$ is the only point of $\mathrm{\partial }{E}_r(P)$ outside it (Heat balls and their time slices).

Representation formula: for $u$ of class $C^{2,1}$ near the closed heat ball $E_r(P)$, $u(P)={\iint }_E(\mathrm{\Gamma }(x_P-y,t_P-s)-r^{-n})(u_t-\mathrm{\Delta }u)+{\mathrm{\Lambda }}_r(P)[u],$ where ${\mathrm{\Lambda }}_r(P)[g]:=\frac{1}{2r^n}{\int }_0^{r^2\mathrm{/}4\pi }\frac{{\rho }_n(\tau )}{\tau }{\int }_{\mathrm{\mid }y-x_P\mathrm{\mid }={\rho }_n(\tau )}g(y,t_P-\tau )dS(y)d\tau$ for continuous $g$; applied to the constant function $1$, whose forcing vanishes, it gives ${\mathrm{\Lambda }}_r(P)[1]=1$ (Heat-ball representation formula).

The lateral functional is a positive measure with density $R(\tau {)}^n\mathrm{/}(2r^n\tau )$ in the sphere-time parametrization for $0<\tau <r^2\mathrm{/}(4\pi )$, as computed in Heat-ball representation formula. Its total mass is one by [F4]. Every nonempty open piece of this parameter domain has positive measure: for $n\ge 2$, a regular sphere chart has strictly positive Gram density, and a small coordinate box has positive Lebesgue measure; for $n=1$ each of the two sphere points has counting mass one. Thus a continuous nonnegative function with zero lateral integral vanishes on the parametrized part of $L_r(P)$. It also vanishes at the lower tip by continuity, since that tip is a limit of lateral points. The lateral level itself is not compact (it omits the top); no compactness assertion or mass at a tip is needed (Surface integration on compact C1 hypersurfaces, A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included, Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative integral agrees with the simple integral on simple functions).

Chaining: for open $\mathrm{\Omega }$, times $0<t_1<t_0\le T<\mathrm{\infty }$ and a Lipschitz path $\gamma :[0,1]\to \mathrm{\Omega }$ with compact image $K$, there is $\rho >0$ with $E_{\rho }(P)\subseteq \mathrm{\Omega }\times (0,T]$ for every $P\in K\times [t_1,t_0]$, and for every $s\in [t_1,t_0)$ there is a chain $P_0=(\gamma (0),t_0),\dots ,P_N=(\gamma (1),s)$, all lying in $K\times [t_1,t_0]$, with $P_{j+1}\in E_{\rho }(P_j)$ (Heat-ball chains reach earlier points).

${\mathrm{\Omega }}_0$ is open and connected, hence polygonally connected: any two of its points are joined by a polygonal, hence Lipschitz, path with compact image (Every connected component of an open subset of ${\mathbb{R}}^n$ is open and polygonally connected, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

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 $\delta$ serving every point); $\bar{Q}$ is compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ 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) and $u\in C(\bar{Q})$.