A\,\,\,\, \Rightarrow\,  {T_{*}}(u_{0}) = +\infty   \bigr\}.$$ But unfortunately, such a point of view makes no sense, owing to the meaning of $({T_{*}}(u_{0}) - t)$ when ${T_{*}}(u_{0}) ~= +\infty$. We have to proceed in an other way and it may be removed by defining a new object $M^{\sigma_{s}}_{c}$ $$M^{\sigma_{s}}_{c}  \eqdefa  \inf_{\substack{u_{0} \in \dot{H}^s \\ {T_{*}}(u_{0}) < \infty    }}   \bigl\{  \limsup_{t \to {T_{*}}(u_{0})}({T_{*}}(u_{0}) - t)\, \| NS(u_{0})(t) \|^{\sigma_{s}}_{\dot{H}^s} \bigr\}.$$ Clearly, (\[remark c\]) implies that $M^{\sigma_{s}}_{c} $ exists and is positive. As we have decided to work under hypothesis $\mathcal{H}$, *a fortiori*, this implies that $M^{\sigma_{s}}_{c} $ is finite. The definition below is the key notion of critical solution in this context.

*[(Sup-critical solution)\
Let $u_{0}$ be an element in $\dot{H}^s$. We say that $u = NS(u_{0})$ is a sup-critical solution if $NS(u_{0})$ satisfies the two following assumptions: $${T_{*}}(u_{0}) < \infty \quad \hbox{and} \quad  \limsup_{t \to {T_{*}}(u_{0})} ({T_{*}}(u_{0}) - t)\,  \| NS(u_{0})(t) \|^{\sigma_{s}}_{\dot{H}^s} =\, M^{\sigma_{s}}_{c}.$$ ]{}*

A natural question is to know if such elements exist. The statement given below gives an affirmative answer and provides a general procedure to build some sup-critical solutions. Our main result follows.

*[(Key Theorem)\
\[Big key theorem\] Let us assume that there exists $u_{0}$ in $\dot{H}^s $ 