  Inspiring by a recent work [57], we analyse a 3-wave kinetic equation,
derived from the elastic beam wave equation on the lattice. The ergodicity
condition states that two distinct wavevectors are supposed to be connected by
a finite number of collisions. In this work, we prove that once the ergodicity
condition is violated, the domain is broken into disconnected domains, called
no-collision and collisional invariant regions. If one starts with a general
initial condition, whose energy is finite, then in the long-time limit, the
solutions of the 3-wave kinetic equation remain unchanged on the no-collision
region and relax to local equilibria on the disjoint collisional invariant
regions. The equilibration temperature will differ from region to region. This
behavior of 3-wave systems was first described by Spohn in [55], without a
detailed rigorous proof. Our proof follows Spohn's physically intuitive
arguments.
