Duhamel solution for a time-independent source

The constant curve $s\mapsto f$ belongs to $C([0,T];L^p({\mathbb{R}}^n))$, so [F1] applies to it: $Df\in C([0,T];L^p({\mathbb{R}}^n))$, $Df(0)=0$, and $Df$ satisfies the forced semigroup relation with the constant forcing.

The two representations agree. For each fixed $t$ the curves $s\mapsto H_{t-s}f$ and $\tau \mapsto H_{\tau }f$ are norm continuous by [F2], and they are related by the reflection $s\mapsto t-s$ of $[0,t]$; by [F3] both Bochner integrals are limits of the integrals of simple approximations, and for a finitely-valued approximation the substitution reduces to the equality of the Lebesgue measures of a measurable level set and its reflection, so passing to the limit gives ${\int }_0^t{H}_{t-s}fds={\int }_0^t{H}_{\tau }fd\tau$.

Classical compactly supported case. If $f$ is spatially Hölder continuous with compact support, then as a function of $(x,t)$ it is bounded, jointly continuous and uniformly spatially Hölder on ${\mathbb{R}}^n\times [0,T]$; by [F1] the scalar potential $u(t,x)={\int }_0^t\int \mathrm{\Gamma }(x-y,t-s)f(y)dyds$ is $C^{1,2}$ with $u_t-\mathrm{\Delta }u=f$ and $u(0,\cdot )=0$, and it represents the Bochner potential $Df$ in the sense of [F2]; moreover $\mathrm{\mid }u\mathrm{\mid }\le T\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, so $u$ lies in the Gaussian growth class and is the unique classical solution with zero initial data there. Hence $(Df{)}_t-\mathrm{\Delta }Df=f$ on ${\mathbb{R}}^n\times (0,T]$, the representation of step 2.1 shows that the two displayed formulas for $Df$ coincide, and no commutation of the unbounded Laplacian with the flow on arbitrary $L^p$ data is asserted.