Absence of singularities of Gaussian beams in diffusion equation case

V. M. Babich

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 21, Tome 195 (1991), pp. 14-18

Résumé

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.