Duhamel formula for the heat equation

Statement

For smooth rapidly decaying initial data $u_0$ and a smooth source $f(t,x)$ with a common compact spatial support on a finite time interval, a solution of

\partial_tu-\nu\Delta u=f,\qquad u(0)=u_0,

is given by the heat Duhamel formula

u(t)=T_tu_0+\int_0^t T_{t-s}f(s)\,ds,

where $T_t$ is the heat semigroup.

Verification and scope

Differentiating the integral gives the endpoint value $f(t)$ ; its remaining terms give $\nu\Delta\int_0^t T_{t-s}f(s)\,ds$ . For these smooth data, spatial derivatives may instead fall on $f$ , justifying differentiation near $s=t$ . The integral vanishes at time zero. Rougher sources require the corresponding convergence and regularity estimates before the formula is interpreted as a classical solution.