to extractions, we get the statements below\
$\bullet$ the profile decomposition of such a sequence of data has a unique profile $\varphi^{j_{0}} $ with constant scale such that $NS(\varphi^{j_{0}} )$ is a sup-critical solution which blows up at time $1$, e.g $$\label{prop point 1}
\limsup_{\tau \to 1} (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} =\, M^{\sigma_{s}}_{c}.\\$$ $\bullet$ “The limsup is actually a sup” $$\label{prop point 2}
\sup_{\tau < 1} (1 - \tau)\,  \| NS(\varphi^{j_{0}})(\tau) \|^{\sigma_{s}}_{\dot{H}^s} \,  =\, M^{\sigma_{s}}_{c}.\\$$ ]{}*

Let $(v_{0,n})_{n \geqslant 1}$ be a bounded sequence in $\dot{H}^s $, satisfiying the assumptions of Proposition \[proposition critical element \]. Therefore, $(v_{0,n})_{n \geqslant 1}$ has the profile decomposition below $$\begin{split}
 v_{0,n}(x) &= \sum_{\stackrel{j \in \mathcal{J}_{1}}{j \leqslant J}} \varphi^{j}(x-x_{n,j})
           + \sum_{\stackrel{j \in \mathcal{J}^{{}{c}}_{1}}{j \leqslant J}} \Lambda^{\frac{3}{p}}_{\lambda_{n,j},x_{n,j}}\varphi^{j}(x) + \psi_{n}^{J}(x).
 \end{split}$$ We denote by ${\displaystyle}{ \tau^{*}_{j_{0}} \eqdefa \inf_{j \in \mathcal{J}_{1}}{{T_{*}}(\varphi^{j})}}$.\
$\bullet$ Step $1$ : we start by proving by a contradiction argument that $\tau^{*}_{j_{0}} =1$.\
\
We have already known by vertue of Lemma \[lemme allure de la solution\], that $\tau^{*}_{j_{0}} \leqslant1$. Assuming that $\tau^{*}_{j_{0}} <1$, we expect a contradiction. Moreover, orthogonal Estimate (\[Pythagore\]) can be bounded from below by $$\label{equation de reference 2}
\begin{split}
\| NS(v_{0,n})(\tau) \|^2_{\dot{H}^s} \geqslant \| NS(\varphi^{j_{0}})(\tau) \|^2_{\dot{H}^s}  - |\gamma_{n}^{J}(\tau)|.
\end{split}$$ 