Fundamental solution for a parabolic PDE with constant coefficents [Cross posting https://math.stackexchange.com/questions/1374384/fundamental-solution-for-a-parabolic-pde-with-costant-coefficents ]
I don't know if this question is more appropriate in Mathematics and not here, in this case I will delete it.
As it is well known, the fundamental solution of the heat equation is the function
$G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x|^2}{4t}}$,
for all $t>0,x\in\mathbb{R}^n$.
I wonder if exists (and if you have same references) a similar explicit formula for the fundamental solution for a parabolic PDE with constant coefficents.
It is possible that it can be found in "Linear and quasilinear equation of parabolic type" by Ladyzenskaja, Solonnikov and Ural'ceva, but as I cant' consulte the book by now, I don't know.
 A: For an operator of the form $L=\partial_t-\sum A_{jk}\partial_j\partial_k$, the fundamental solution is computed in Section 3.3 of Volume I of Hormander's treatise (The Analysis of Linear PDOs). I think one may try to extend the formula to include lower order terms by playing with it a little; for instance if $u$ solves $Lu=0$, then $v = e^{-at}u$ solves $Lv+av=0$, and similarly $v=(a_1x_1+\dots+a_nx_n)u$ solves an equation with additional first order terms.
A: Yes, it can be written out explicitly. Namely, for the operator
$$
\partial_t-\sum_{i,j=1}^na_{ij}\partial_{ij}-\sum_{i=1}^nb_i\partial_i-c,
$$
where $A=(a_{ij})>0$, $b=(b_1,\ldots,b_n)$,
$$
G(t,x)=\frac{1}{(4\pi t)^{n/2}|A|^{1/2}}\exp\left\{-\frac{(A^{-1}(x-bt),x-bt)}{4t}+ct\right\}.
$$
It can be obtained either via the Fourier transform wrt $x$ as in the case of the heat equation or making an affine change of variables in $\mathbb R^{n}$, reducing again to the heat equation.
A: A good place to start is the series of articles by P. Wagner and N. Ortner.  In addition to their constructive proof of the Malgrange-Ehrenpreis theorem on the existence of fundamental solutions for pde's with constant coefficients (AMS 116---MR 2510844), they have, in a series of previous papers, computed explicit solutions for many concrete examples.
