Absence of singularities of Gaussian beams in diffusion equation case

The diffusion equation in the case of point source is considered:
$$ \varepsilon\frac1h\frac{\partial}{\partial x^i}\left(D^{ij}h\frac{\partial c}{\partial x^j}\right)-U^i\frac{\partial c}{\partial x^i}=-A\delta(x-x_0),\quad x=x^1,\dots,x^m,\quad x_0=x_0^1,\dots,x_0^m, $$
where $\varepsilon$ is a small parameter. The asymptotic expansion of $c$ reduces to Gaussian beam solution concentrated in a small neighbourhood of the curve $l$, which is solution of the system of differential equation:
$$ \frac{d}{d\,s}x^i=U^i,\quad x^i\mid_{s=0}=x_0^i. $$
Absence of singularities of Gaussian beams is proved.