Duhamel's principle and forcing

Let S(t) be the heat semigroup for a specified spatial domain and homogeneous boundary condition. For uₜ=κΔu+f with initial state u₀, the formula below adds the freely evolved initial state to source contributions S(t−s)f(s). On the whole space, each term is a Gaussian convolution in space. On a bounded Dirichlet domain or a periodic torus, S is different. The elapsed time t−s is essential; using S(t) for every source slice would incorrectly diffuse recently added heat for the full observation time.

For u₀ in L² and f integrable in time with values in L² on a finite interval, the Bochner integral defines a continuous L² mild solution. It satisfies the evolution equation in an appropriate weak or distributional sense, but f of this regularity need not make u classically differentiable at every point. Additional regularity supports differentiating the formula and recovering a classical PDE. Calling the integral a solution is meaningful only after specifying which of these senses is intended.

If a spatial eigenfunction has negative Laplacian eigenvalue −λ, its coefficient obeys c′+κλc=g(t). Variation of constants gives an exponentially weighted time integral. A constant source produces a transient plus a stationary coefficient when λ>0. A zero mode behaves differently: with no damping, its coefficient accumulates the time integral of the source mean. Whether that mode exists depends on the boundary condition. A periodic or insulated problem may heat up indefinitely under a positive mean source even when every nonconstant mode remains bounded.

If g(t) has the same exponential decay as the homogeneous mode, the integral produces a factor t. This is sometimes called temporal resonance in the coefficient calculation, but it does not imply unbounded growth: t times a decaying exponential still tends to zero. Formulas obtained by dividing the difference of two exponential rates must be replaced by their limit when the rates coincide. Evaluating the scalar integral directly is often safer than memorizing a quotient whose denominator can vanish.

A semigroup built for zero Dirichlet data cannot directly propagate arbitrary nonzero boundary values. One method is to choose a sufficiently regular lifting h with the required boundary trace and write u=v+h. Then v has homogeneous boundary data and an effective source f−hₜ+κΔh, together with adjusted initial data. The lifting's regularity and compatibility at time zero must be checked. Ignoring those terms solves a different boundary problem even if the interior source integral looks familiar.

For a semilinear equation, replace f by a function of the unknown solution inside the integral. The resulting expression is an integral equation used in a fixed-point argument, not an explicit formula. Local Lipschitz control on bounded ranges can produce local existence and uniqueness, but does not automatically prevent later blow-up. Likewise, an a priori source estimate cannot be inserted before establishing it for the evolving solution. Duhamel organizes the analysis; it does not remove the nonlinear estimates or the need to identify a continuation criterion.