Nonhomogeneous heat equation solution of $u_t - \Delta u = \rm div f$

Question

For 3D, I have some doubts of the nonhomogeneous heat equation with initial value zero. i.e if $u_t - \Delta u = f$, $$u(x,t)=c\int_{0}^{t} \int_{\mathbb R^3} \frac{1}{(t-s)^{3/2}} e^{\frac{-|x-y|^2}{4(t-s)}}f(y,s)dyds$$.

Now if I change $f$ to $\rm div f$, here I mean the divergence. Then, what will $u$ be? Thanks.

Answer 1

We just need to apply integration by parts to get rid of the $div$, i.e. the $div$ will become the gradient on the heat kernel.