  The self-similar growth-fragmentation equation describes the evolution of a
medium in which particles grow and divide as time proceeds, with the growth and
splitting of each particle depending only upon its size. The critical case of
the equation, in which the growth and division rates balance one another, was
considered by Doumic and Escobedo in the homogeneous case where the rates do
not depend on the particle size. Here, we study the general self-similar case,
using a probabilistic approach based on L\'evy processes and positive
self-similar Markov processes which also permits us to analyse quite general
splitting rates. Whereas existence and uniqueness of the solution are rather
easy to establish in the homogeneous case, the equation in the non-homogeneous
case has some surprising features. In particular, using the fact that certain
self-similar Markov processes can enter $(0,\infty)$ continuously from either
$0$ or $\infty$, we exhibit unexpected spontaneous generation of mass in the
solutions.
