[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.



Fundamental solution for a parabolic PDE with constant coefficents
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.


