Non-uniqueness of Leray-Hopf solutions for a dyadic mode

The dyadic model $\dot u_n + \lambda^{2n}u_n - \lambda^{\beta n}u_{n-1}^2 + \lambda^{\beta(n+1)}u_nu_{n+1} = f_n$,$u_n(0)=0$, is considered. It is shown that in the case of non-trivial right hand side the system can have two different Leray-Hopf solutions.