q}$, with $3<p,q<\infty$, the $NS$-solution, (the lifespan of which is assumed finite) becomes unbounded at the blow-up time. Let us say a few words about the limit case $\dot{B}^{-1+ \frac{3}{p}}_{p, \infty}$. We may wonder if the result holds in the limit case $q=\infty$. As far as the author is aware, the answer is still open. Actually, if it holds, *a fortiori* it holds in the smaller space $\dot{B}^{\frac{1}{2}}_{2, \infty}$, by vertue of the embedding ${\displaystyle}{\dot{B}^{\frac{1}{2}}_{2, \infty}  \hookrightarrow \dot{B}^{-1+ \frac{3}{p}}_{p, \infty}}$. In others terms, it would mean there is no blowing-up solution, bounded in the critical space $\dot{B}^{\frac{1}{2}}_{2, \infty}$. This is related to the concern of our paper since we build some blowing-up solutions bounded in this critical space, under the assumption of blow up at minimal rate. We mention the very interesting work of H. Jia and V. $\breve{\mathrm{S}}$ver$\acute{\mathrm{a}}$k [@JV], where they prove that $-1$-homogeneous initial data generate global $-1$-homogeneous solutions. Unfortunately, the uniqueness of such solutions is not guaranteed.

Existence of sup-critical solutions
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0.5cm The goal of this section is to give a partial proof of Key Theorem \[Big key theorem\]. It relies on the two Lemmas below.

*[(Existence of sup-critical solutions in $\dot{H}^s $)\
\[general lemma for 