and $M$ in $\R^{+}_{*}$ such that $${T_{*}}(u_{0}) < \infty \quad \hbox{and} \quad   ({T_{*}}(u_{0}) - t)\,  \| NS(u_{0})(t) \|^{\sigma_{s}}_{\dot{H}^s} \leqslant M.$$ Then, there exists $\Phi_{0} \in \dot{H}^s \cap \dot{B}^{\frac{1}{2}}_{2,\infty}$ such that $\Phi \eqdefa NS(\Phi_{0})$ is a sup-critical solution, blowing up at time $1$, such that $$\label{key theorem point 1}
\sup_{ \tau < 1} \,(1 - \tau)\,  \| NS(\Phi_{0})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} =\,  \limsup_{\tau\to 1} (1 - \tau)\,  \| NS(\Phi_{0})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} =\, M^{\sigma_{s}}_{c}.$$ In addition, there exists a positive constant $C$ such that $$\label{key theorem point 2}
\hbox{and for any} \quad \tau <1, \quad \| NS(\Phi_{0})(\tau) \|_{\dot{B}^{\frac{1}{2}}_{2,\infty}} \leqslant C,$$ where the Besov norm (for regularity index $0<\alpha <1$) is defined by $$\| u \|_{\dot{B}^{\alpha}_{2,\infty}} \eqdefa \sup_{x \in \R^d} \,\, \frac{\| u(\cdotp -x) - u\|_{L^{2}}}{|x| ^{\alpha}}\cdotp$$ ]{}*

We postpone the proof of (\[key theorem point 1\]) of the Key Theorem \[Big key theorem\] to the next section. The proof of (\[key theorem point 2\]) will be given in Section $5$. We stress on the fact that (\[key theorem point 2\]) is somewhat close to a question raised by the paper of I. Gallagher, G. Koch and F. Planchon [@GKP2], in which they prove that for any initial data in the critical Besov space $\dot{B}^{-1+ \frac{3}{p}}_{p, 