  Markovian growth-fragmentation processes describe a family of particles which
can grow larger or smaller with time, and occasionally split in a conservative
manner. They were introduced in a work of Bertoin, where special attention was
given to the self-similar case. A Malthusian condition was notably given under
which the process does not locally explode, in the sense that for all times,
the masses of all the particles can be listed in non-increasing order. Our main
result in this work states the converse: when this condition is not verified,
then the growth-fragmentation process explodes almost surely. Our proof
involves using the additive martingale to bias the probability measure and
obtain a spine decomposition of the process, as well as properties of
self-similar Markov processes.
