A Result on the Blow-up Rate for the Zakharov System in Dimension 3

We consider a blow-up solution (u,n,v) of the Zakharov system in ${\mathbb R}^3$: $$\left\{ \begin{array}{l} iu_t=-\Delta u+nu,\\ n_t=-\nabla\cdot v,\\ v_t=-\nabla (n+|u|^2). \end{array} \right.$$ If T is the finite blow-up time, we show the following integral estimate for n: $$ \int_0^T\left(\int_{{\mathbb R}^3}|n(x,t)|^q dx\right)^{\frac{\g}{q}}dt=+\infty, $$ where $\epsilon\in\ ]0,\frac{1}{4}]$, $%%%%%{\di q=\frac{3}{2(1-\epsilon)}\in\ \left]\frac{3}{2},2\right]$, %$%%%}$ and ${\gamma>\frac{1}{\epsilon}}$. In particular, this implies that, for a < 1$, $${\di \sup\limits_{t\in [O,T)}\left( \left(T-t\right)^{a\epsilon}\left(\int_{{\mathbb R}^3}|n(x,t)|^q dx\right)^{\frac{1}{q}}\right)=+\infty.}$$