the parabolic maximum principle
Heat a metal bar and watch it cool. No interior spot ever spontaneously becomes hotter than the bar was at the start, or hotter than the ends are being held — the hottest temperature you will ever see was already present either in the initial profile or at the boundary. Diffusion only flattens; it never manufactures a fresh peak out of nowhere. The parabolic maximum principle states exactly this for the heat equation.
The subtlety is which part of the boundary counts. For an evolving problem on a region in space across a time interval [0, T], the relevant boundary — called the parabolic boundary — is the bottom (the initial time t = 0) together with the sides (the spatial boundary for all times), but NOT the top (the final time t = T). The principle says: if u_t <= k Laplacian u (a subsolution), the maximum of u over the whole space-time cylinder is attained on this parabolic boundary. So you reach back to the initial data and the side data, never forward to the final slice. The proof again perturbs by a term whose sign forces the extremum onto the parabolic boundary; the asymmetry in time — past matters, future does not — is exactly the irreversibility of diffusion showing up in the geometry.
This delivers everything the elliptic version does, now in time: uniqueness of the heat initial-boundary-value problem, continuous dependence on initial and boundary data, and the comparison principle (a hotter start and hotter boundary give a pointwise hotter solution for all later time). It also explains why the backward heat equation is ill-posed — running the principle backward would let interior peaks grow, which diffusion forbids. There is a strong parabolic version too: a subsolution that attains an interior space-time maximum is constant up to that time.
A rod starts at temperatures between 0 and 100 degrees and has both ends held at 20 degrees thereafter. The parabolic maximum principle guarantees that at every later time, every interior point stays between 0 and 100 — no point can ever exceed the hottest value present at the start or on the ends.
The maximum lives on the parabolic boundary: initial time plus the sides, not the final time.
The top of the cylinder (the final time) is deliberately excluded — that is the whole point. A common slip is to include t = T in the maximizing set; doing so would wrongly suggest you could read the past off the future, which the irreversibility of diffusion (and the ill-posedness of the backward heat equation) flatly forbids.