  Inspired by the fundamental work of Escobedo and Velazquez
\cite{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.
