Blow up of the critical norm for some radial L^2 super critical nonlinear Schrodinger equations

We consider the nonlinear Schr"odinger equation $i{u}_t=-\mathrm{\Delta }u-\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in dimension $N\ge 3$ in the $L^2$ super critical range $1+\frac{4}{N}<p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is ${\dot{H}}^{s_c}$ with $0<s_c<1$ and this covers the physically relevant case $N=p=3$. The existence of finite time blow up solutions is known. Let $u(t)\in \dot{H}^{s_c}\cap \dot{H}^1$ be a radially symmetric blow up solution which blows up at $0<T<+\mathrm{\infty }$, we prove that the scaling invariant $L^{p_c}$ norm where ${\dot{H}}^{s_c}\rightharpoonup L^{p_c}$ also blows up with a lower bound $\mathrm{\mid }u(t){\mathrm{\mid }}_{L^{p_c}}\ge \mathrm{\mid }\log (T-t){\mathrm{\mid }}^{C_{N,p}}$ as $t\to T$.