  We study the four-wave kinetic equation of weak turbulence linearized around
the Rayleigh-Jeans spectrum when the collision integral is associated with
short-range interactions between non-relativistic bosonic quasiparticles. The
technique used for the analysis of the stability is based on the properties of
the Mellin transform of the kernel in the integral equation. We find that any
perturbation of the Rayleigh-Jeans distribution evolves towards low momentum
scales in such a form that, when $t \to \infty$, all the particles occupy a
sphere of radius arbitrary small.
