  This paper addresses the construction and the stability of self-similar
solutions to the isentropic compressible Euler equation. The objects of our
study are smooth and non-smooth Guderley solutions; they model a gas that
implodes isotropically, ending in a singularity formation in finite time. Here,
we provide the first construction of smooth solutions and the first
characterization of the spectrum of radial perturbations as well as some
endpoints of unstable modes. One of these endpoints is a shock formation that
happens before the singularity at the origin, showing that the implosion
process is unstable.
