Let $\Omega=\mathbb{R}^n\times \mathbb{R}^+$ and $a_0,\ldots, a_n\in \mathbb C$, I must find a fundamental solution of the PDE $$(*)\qquad\qquad\frac{\partial u}{\partial t}(x, t)+\sum_{k=1}^n a_k\frac{\partial u}{\partial x_k}(x, t)+a_0 u(x, t)=f(x, t),\qquad (x, t)\in \Omega.$$ If we call $$L=\frac{\partial }{\partial t} +\sum_{k=1}^n a_k\frac{\partial }{\partial x_k} +a_0,$$ then it is enough to find a fundamental solution for $L$, say $T$ (i.e., $LT=\delta$, and afterwards we just convolute with $f$)
How should I proceed here? I know that every non-zero linear differential operator with constant coefficients has a fundamental solution (and one can even show an explicit formula by means of one of the constructive proofs of the Malgrange-Ehrenpreis theorem), but this is rather long and cheap.
Is there an easier(direct) way to find such solution? Any help or reference is highly appreciated
For the computation of the fourier transform method, some progress:
$LT=\delta$ as you've written, and then we can find $T$ via Fourier transform. I'm going to rename your equation slightly,
$u_t + \textbf{a}\cdot \nabla_x u(x,t) + bu(x,t) = f(x,t)$
I take the convention that(as in Evan's PDE book)
$$\mathcal{F}[u](\xi)=\frac{1}{(2 \pi)^{\frac{n}{2}}}\int_{\mathbb{R}^n}u(x)e^{-ix \cdot \xi}\,dx$$
$$\mathcal{F}^{-1}[u](x)=\frac{1}{(2 \pi)^{\frac{n}{2}}}\int_{\mathbb{R}^n}u(\xi)e^{i\xi \cdot x}\,d\xi$$
Then,
$\mathcal{F}[\partial_t u]=\hat{u}_t$
$\mathcal{F}[a \cdot \nabla u] = i(a \cdot \xi) \hat{u}$
$\mathcal{F}[\delta(x)] = 1$
Hence, we arrive to the equation
$$\hat{T}_t + (i(a \cdot \xi)+b)\hat{T} = 1$$
so the Fourier transform of the fundamental solution satisfies
$$\hat{T}(\xi,t) = C e^{-t(i(a \cdot \xi)+b)} + (i(a \cdot \xi)+b)^{-1}$$
where $C$ is yet to be found. However from here, I've been having some trouble writing out the inverse transform of each of those functions.