On the energy transfer towards large values of wavenumbers for solutions of 4-wave kinetic equations

Inspired by the fundamental work of Escobedo and Velazquez {EscobedoVelazquez:2015:FTB,EscobedoVelazquez:2015:OTT}, we prove that solutions of 4-wave kinetic equations, under very general forms of the dispersion relations, exhibit the transfer of energy towards large values of wavenumbers as time evolves. We also provide a global well-posedness theory for mild solutions of the equation under general forms of the dispersion relations.