Where guide 1 left us
In the previous guide we stopped thinking of u(x, t) as a surface over space-time and started thinking of it as a moving point: at each instant t, the state u(t) is a whole function of x living in a function space, and the equation u_t = -A u tells that point which direction to move. Here A is a parabolic operator — the model is A = -Laplacian, so the heat equation u_t = Laplacian u becomes the abstract evolution problem u_t + A u = f. We never solved it; we only reframed it. This guide actually catches a solution and then squeezes it.
Just as in the elliptic rung, the strategy is to refuse to differentiate twice. Multiply by a test function v, integrate by parts once in space, and the spatial part collapses into the same bilinear form B(u, v) you already met for elliptic problems. What is new is time: a weak solution of the parabolic problem is a curve t goes to u(t) so that, for every test v, the time-derivative of the inner product of u with v, plus B(u(t), v), equals L(v) — an ordinary differential equation in disguise, one for each test function at once.
Catching a solution: Galerkin in time
The trick that produced an elliptic solution works again, with one twist. We pick a growing list of basis functions in space — eigenfunctions of A are the classic choice — and demand the weak equation only against those finitely many directions. In the elliptic case that left us a matrix equation; here it leaves us a small system of ordinary differential equations for the time-dependent coefficients. That system is linear with smooth coefficients, so the existence theory for ODEs hands us an approximate solution u_m(t) for free, on every time interval. This is the Galerkin method, now running in time.
But an approximation is worthless unless we can stop it from blowing up. The whole game now is to bound u_m uniformly in m, so that the approximations stay trapped in a fixed region of the function space and a subsequence can converge. That uniform leash is the energy estimate of the next section — and the beautiful thing is that we extract it by the most natural move imaginable: test the equation against the solution itself.
The energy estimate: testing against u itself
Take the heat equation u_t = Laplacian u, multiply by u, and integrate over space. The left side becomes one half the time-derivative of the integral of u^2 — the rate of change of the total energy E(t) = the integral of u^2. The right side, after integrating by parts, becomes minus the integral of |gradient u|^2, which is never positive. So we land on a single honest sentence: the energy only ever goes down, and it falls at exactly the rate that the gradient is steep. This is the engine of the whole existence proof, and it is one line.
Multiply u_t = Lap u by u, integrate over the domain:
d/dt ( 1/2 * integral u^2 ) = integral (u * Lap u)
= - integral |grad u|^2 (<= 0)
So E(t) = integral u^2 is non-increasing:
E(t) <= E(0) for every t > 0
The gradient term you discard is exactly the dissipation
-- the heat the system is shedding as it smooths.Run that same computation on each Galerkin approximation u_m and the bound holds for every m with the same constant — because the right side was discarded, never used. That is precisely the uniform leash we needed. The approximations cannot escape; the energy of every one of them is capped by the energy of its initial data. A bounded sequence in the right function space has a weakly convergent subsequence, its limit satisfies the weak equation, and that limit is the weak solution. Notice we built it without ever writing a formula — this is an a priori estimate, a bound assumed of the solution before we are sure the solution exists, which then forces the solution into being.
The maximum principle: bounds without solving
The energy estimate controls an average — the integral of u^2. The parabolic maximum principle controls something far sharper: the actual pointwise size of u. Its statement is almost physical common sense. For u_t = Laplacian u on a rod over a time interval, the maximum of the temperature is attained on the parabolic boundary — the initial time slice together with the two ends across all time. In plain words: heat can only get as hot as it started or as the boundary forces it; an interior point in the future can never spontaneously become hotter than everything that fed it.
Why is it true? At an interior maximum in space, the function curves downward, so Laplacian u is less than or equal to zero, which forces u_t less than or equal to zero — the hottest point is already cooling, so a new record high cannot be born in the interior. That is the weak maximum principle, which allows the maximum to merely be attained on the parabolic boundary. The strong version says more: unless u is constant, the maximum is attained ONLY there, never touched again in the interior. This is the same downhill logic you saw for harmonic functions in the elliptic rung, now carrying a time arrow.
From this one principle, three deep facts fall out almost for free. First, uniqueness: if two solutions share the same initial and boundary data, their difference solves the heat equation with zero data, so its maximum and minimum are both zero — they are identical. Second, comparison: the comparison principle says if one solution starts below another and stays below on the boundary, it stays below forever; order is preserved by the flow. Third, a quiet but profound consequence — the heat equation respects bounds it never had to be told about, which is the analytic face of physical irreversibility.
Honest fine print
Three honesties before you climb on. First, the time arrow is not decoration — it is structural. The energy estimate and the maximum principle both lean on the sign of u_t, and that sign only points the safe way forward in time. Run the heat equation backwards and the energy would have to grow without bound and the maximum principle reverses; this is why the backward heat equation is famously ill-posed. Existence here means existence for t increasing, and that asymmetry is the whole reason diffusion is irreversible — the irreversibility of diffusion is baked into the same estimates that give existence.
Second, the maximum principle is a gift of LINEARITY and the right sign of the operator. It holds beautifully for u_t = Laplacian u and for linear parabolic operators with the correct ellipticity, and it extends to many reaction-diffusion equations through comparison with carefully chosen sub- and super-solutions. But it is not a universal law of all evolution equations: a system of equations, or the wrong sign, can destroy it entirely. Do not assume a maximum principle; check that your operator earns one.
Third, mind the gap between weak and classical, exactly as in the elliptic rung. The Galerkin limit is only a weak solution — it satisfies the integrated equation, not yet the pointwise PDE. That it is in fact smooth for t greater than zero is a separate miracle, the instantaneous smoothing that the next guide is devoted to. Existence first, energy and maximum bounds second, smoothness third: keeping those three questions apart is the cleanest way to climb this rung.