Duhamel's formula
/ doo-ah-MELL /
Suppose your evolution equation has a forcing term — heat is being added continuously by a source, not just left to diffuse from an initial profile. Duhamel's idea is beautifully simple: treat the source as a stream of tiny instantaneous 'kicks', let each kick evolve on its own from the moment it is delivered, and add up all their contributions. The total response at time t is the sum (integral) over all earlier times s of 'the kick delivered at s, then propagated forward for the remaining time t - s'. It is the same variation-of-constants idea you met for linear ODEs, lifted to the semigroup setting.
Concretely, for the inhomogeneous abstract Cauchy problem du/dt = -A u + f(t), u(0) = u_0, where -A generates a semigroup e^{-tA}, Duhamel's formula reads: u(t) = e^{-tA} u_0 + integral from 0 to t of e^{-(t-s)A} f(s) ds. Read the two pieces literally: e^{-tA} u_0 is the free evolution of the initial data (the homogeneous solution), and the integral superposes the source — at each past instant s the input f(s) is injected and then carried forward by the semigroup over the elapsed time t - s. A solution defined by this formula (rather than by being differentiable and satisfying the equation pointwise) is called a mild solution, and Duhamel's formula is exactly its definition.
Its real power shows up for nonlinear (semilinear) equations du/dt = -A u + F(u). You cannot solve those by hand, but you can rewrite them as the fixed-point equation u(t) = e^{-tA} u_0 + integral from 0 to t of e^{-(t-s)A} F(u(s)) ds and solve by contraction mapping: feed in a guess for u, compute the right-hand side, repeat. When the semigroup is analytic, its smoothing estimate norm of e^{-(t-s)A} <= C with the 1/sqrt control on derivatives makes the integral operator a contraction on a short time interval, giving local existence and uniqueness — the standard route to local well-posedness for reaction-diffusion and many other parabolic models.
For the heat equation with a source, u_t = Laplacian u + f(x, t), u(., 0) = u_0, Duhamel gives u(t) = e^{t Laplacian} u_0 + integral from 0 to t of e^{(t-s) Laplacian} f(s) ds. Concretely each piece is a convolution against the heat kernel: the free part blurs u_0 over time t, and the forced part blurs each f(., s) over the shorter remaining time t - s and integrates.
The forced solution = free evolution of the data + accumulated propagated source.
Duhamel's formula always makes sense as an integral, but a mild solution it produces need not be a classical (differentiable) solution unless the data and forcing are regular enough — mildness is genuinely weaker, and bridging back to a classical solution requires extra smoothness, which for parabolic problems the smoothing of an analytic semigroup conveniently supplies.