for critical element \], we have to prove that there exists a unique profile with a lifespan $\tau^{*}_{j_{0}} =1$, namely $|\tilde{\mathcal{J}_{1}} | = 1$. Once again, we assume that there exists at least two profiles in $\tilde{\mathcal{J}_{1}}$. We expect a contraction. Arguments of the proof are similar to the ones used in the step $2$. We shall use the fact ${\displaystyle}{(1-\tau)^{\frac{2}{\sigma_{s}}}\, \| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s}}$ can not be small as we want, by vertue of (\[remark c\]). Indeed, let us come back to Inequality (\[equation de reference\]). We have already proved that $ \varphi^{j_{0}}$ generates a sup-critical solution, blowing up at time $1$. It means that for any $\varepsilon >0$, there exists a time $\tau_{\varepsilon}$ such that $$0<  ( 1 -\tau_{\varepsilon} )^{\frac{2}{\sigma_{s}}}\, < \varepsilon \quad  \hbox{and} \quad M_{c}^2 - \varepsilon   \leqslant(1 -\tau_{\varepsilon} )^{\frac{2}{\sigma_{s}}}\,  \| NS(\varphi^{j_{0}})(\tau_{\varepsilon}) \|^{2}_{\dot{H}^s} \leqslant M_{c}^2 + \varepsilon.$$

Therefore, Inequality (\[equation de reference\]) becomes at time $\tau_{\varepsilon} $ $$M^2_{c} + \varepsilon_{n}  \geqslant  M_{c}^2 - \varepsilon \, +\, \sum_{j \in \tilde{\mathcal{J}_{1}}, j \neq j_{0}} (1-\tau_{\varepsilon})^{\frac{2}{\sigma_{s}}}\, \| NS(\varphi^{j})(\tau_{\varepsilon}) \|^{2}_{\dot{H}^s} \, -\,   |\gamma_{n}^{J}(\tau_{\varepsilon})|.$$

By vertue of (\[remark c\]), there exists a universal constant $c >0$ such that for any $j \in \tilde{\mathcal{J}_{1}}$ and $j \neq j_{0}$ $$(1-\tau)^{\frac{2}{\sigma_{s}}}\, \| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s} \geqslant c^2.$$ 