Consider the problem (Evans, Ch 2, 14) $$ u_t-\Delta u+cu=f ,x \in \mathbb R^n\times (0,\infty)$$ $$ u=g , \mathbb R^n\times {t=0} $$
If $u$ solves $ u_t-\Delta u=f$, $u=0$ on and $v$ solves $u_t-\Delta u=0$, then we can use superposition to get a solution to $u_t-\Delta u=f$,$u=g$ on bdy. However, it seems the $cu$ term prevents me from doing this.
Hint: Try using the Fourier transform to solve this PDE. Let $$ \hat{f}(\xi)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^n}e^{-ix\cdot\xi}f(x)dx,~f\in L^1(\mathbb{R}^n),~\xi\in\mathbb{R}^n. $$ Then we can write $$ \hat{u}_t-\xi^2\hat{u}+c\hat{u}=\hat{f} \\ \Rightarrow \hat{u}_t=(\xi^2-c)\hat{u}+\hat{f}. $$ Deal with the cases $n=1$ and $n>1$ separately.