Switching the heater on: Duhamel's formula
By guide 4 you can solve the unforced problem du/dt = -A u, u(0) = u_0: its answer is the single object u(t) = e^(-tA) u_0, the analytic semigroup gliding the initial profile forward. But almost every real parabolic equation carries a source term — a heater pumping warmth into the bar, a chemical reaction creating substance, a forcing f(t) that is itself a moving profile. The question of this final guide is simple to state: how does the same flow respond when something keeps pushing on it? The answer, Duhamel's formula, is one of the most reusable ideas in all of analysis, and you have in fact already met its baby version — the Duhamel principle for the heat equation back in the heat rung.
The idea is to treat the forcing as a continuous bombardment of tiny initial kicks. Imagine the interval from 0 to t chopped into slivers. In the sliver around an earlier time s, the source f(s) deposits a small amount of new profile, f(s) ds, into the bar. From that instant on, this freshly deposited bump simply rides the free flow — it gets carried by e^(-(t-s)A), aged for the remaining time t - s. The full answer at time t is then just the sum, over all earlier instants s, of these aged deposits, plus the original data also aged the whole way. Replace the sum by an integral and you have it exactly.
free flow only: u(t) = e^(-tA) u_0
with forcing f: u(t) = e^(-tA) u_0 + integral from 0 to t of e^(-(t-s)A) f(s) ds
\__________/ \_______________________________________/
data, aged t each kick f(s) ds, aged the leftover time t-s
check t = 0: the integral is empty => u(0) = u_0 (data recovered)The mild solution: an equation that asks for almost nothing
Here is the move that makes the whole modern theory run. Stop treating Duhamel's formula as a consequence of solving the PDE, and start treating it as the definition of the solution. A profile u(t) that satisfies the integral identity u(t) = e^(-tA) u_0 + (the Duhamel integral) for every t is called a mild solution of the abstract Cauchy problem. Notice what this definition does not demand: it never differentiates u in time, and never applies the unbounded A to u directly. Everything on the right-hand side is built only from the semigroup e^(-tA), which is a perfectly nice bounded operator. The roughness has been swept out of sight.
Why is that worth a special name? Because the classical notion — a u(t) that is genuinely differentiable in time and lands in the domain D(A) at every instant — is fragile. It can fail to exist when the data u_0 is merely an L^2 profile rather than a smooth one, or when the forcing f is rough in time, even though the physics is perfectly sensible. The mild solution always exists and is unique under very mild hypotheses (a continuous f and any u_0 in the space X suffice), so you get a solution to hold onto first, and only afterwards ask how smooth it secretly is. Existence is cheap; regularity is the reward you earn separately.
Where analyticity pays its rent: forced smoothing
Now collect the dividend from guide 4's hardest fact — that for a parabolic A the semigroup is not merely a C_0 semigroup but an analytic semigroup, generated by a sectorial operator. Concretely this means e^(-tA) does not just move profiles around; for any t > 0 it pours them into the domain of A and beyond, with sharp control on how much derivative it manufactures: the size of A e^(-tA) blows up only like 1/t as t shrinks to 0. That single estimate is the engine. Feed it through the Duhamel integral and you discover that even a merely continuous, non-differentiable forcing f produces a u(t) that is differentiable in time and sits in D(A) for every t > 0. The flow smooths the forcing the very instant it arrives.
This is the same one-way magic that defines parabolic equations, now seen through the forcing. A rough initial profile, or a rough source, has all its jaggedness erased for t > 0 — the bargain you struck in the mild solution (cheap existence, no smoothness demanded) is repaid in full a heartbeat later, when the solution turns out smooth after all. And it is the same magic that makes the backward heat equation hopeless: smoothing forward means information is irreversibly thrown away, so there is no honest way to run the integral in reverse. The Duhamel formula only ever integrates the semigroup forward, for the elementary reason that backward is where the universe of parabolic problems forbids you to go.
Reading the long run: spectral gap and decay
The very same formula, stared at as t grows large instead of small, answers the last question of the rung: where does everything end up? Split the Duhamel answer u(t) = e^(-tA) u_0 + (the integral). For a steady forcing f that does not depend on time, you can guess the eventual resting profile: it is the steady state u_inf solving A u_inf = f, the moment the rate of change du/dt finally stalls to zero. Subtract it off and let v = u - u_inf measure the leftover. The leftover obeys the unforced equation dv/dt = -A v, so v(t) = e^(-tA) v(0). The whole long-time story is therefore decided by one thing: how fast the free semigroup decays.
And that rate has a name you can read straight off the operator. Because A is a positive elliptic operator on a bounded domain, its spectrum is a discrete ladder of positive eigenvalues 0 < lambda_1 < lambda_2 < ... — the same eigenvalues that ran your old Fourier-mode expansion. The semigroup acts on the lambda_k-mode by multiplying it by e^(-lambda_k t), so every mode dies, and the slowest survivor is the lowest one, decaying like e^(-lambda_1 t). The strictly positive bottom eigenvalue lambda_1 is the spectral gap, and it is the exact exponential rate at which any parabolic system relaxes to its steady state. A bigger gap means a faster forget; this is the rigorous heir of the decay of Fourier modes you first met for the bare heat equation.
When the nonlinearity fights back
The hidden gift of the mild formulation is that it barely cares whether the forcing is linear. Replace the external f(s) by a term f(u(s)) that depends on the solution itself, and the Duhamel identity becomes u(t) = e^(-tA) u_0 + integral of e^(-(t-s)A) f(u(s)) ds — now an implicit equation, the unknown sitting on both sides. This is the home of every reaction-diffusion equation: diffusion supplies the smoothing semigroup, the reaction supplies f(u). Solving it is a fixed-point problem, and the same 1/t smoothing estimate that tamed the linear case lets a contraction-mapping argument produce a unique mild solution — at least for a short time. The abstract machinery you built does not flinch at nonlinearity; it simply turns an integral into a fixed-point equation.
But be honest about the words short time. With a nonlinear reaction, long-time behaviour is no longer the tidy decay-to-equilibrium of the linear world. The reaction can keep feeding energy in, defeating the diffusion's urge to relax: some solutions blow up in finite time (the mild solution ceases to exist), while others settle onto a global attractor — a finite-dimensional set of long-term states far richer than a single steady point, supporting fronts, oscillations, even patterns that diffusion alone could never make. The clean exponential decay was a feature of linearity, not of parabolicity. Strip the linearity and the spectral gap governs only the behaviour near an equilibrium, never the global story.
Step back and see what one formula has carried you through. Duhamel turned the unforced flow into a response to any push; reading it as a definition gave the mild solution, cheap to exist and easy to extend to nonlinearities; the analytic semigroup's smoothing repaid that cheapness with instant regularity; and reading the same formula as t grows revealed the spectral gap as the rate of relaxation — together with the honest warning that nonlinearity can replace that relaxation with blow-up or an attractor. That is the whole arc of parabolic theory, from a single moving point in guide 1 to the long-time fate of forced, nonlinear evolution here. One equation, du/dt = -A u + f, read forward, slowly, and all the way to the end.