L^2-stability for the 4-waves kinetic equation around the Rayleigh-Jeans equilibrium

We consider the four-waves spatial homogeneous kinetic equation arising in weak wave turbulence theory. In this talk I will present some new results on the existence and long-time behaviour of solutions around the Rayleigh-Jeans thermodynamic equilibrium solutions. In particular, introducing a cut-off on the frequencies, I will present an $L^2$ stability of mild solutions for initial data close to Rayleigh-Jeans, when the dispersion relation is weakly perturbed around the quadratic one. If time permits, I will discuss a more recent result (joint work with Miguel Escobedo) on stability of radial solutions without the cut-off on the frequencies.