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

We consider the nonlinear Schr\"odinger equation iuₜ=-Δ u-|u|ᵖ⁻¹u in dimension N≥ 3 in the L² super critical range 1+4/N< p<N+2/N-2. The corresponding scaling invariant space is Ḣsc with 0< sc<1 and this covers the physically relevant case $N=p=3$. The existence of finite time blow up solutions is known. Let pc=N/2(p-1) so that Ḣsc⊂ Lpc. Let u(t)∈ Ḣsc∩ Ḣ¹ be a radially symmetric blow up solution which blows up at 0<T<+∞, we prove that the scaling invariant Lpc norm also blows up with a lower bound: $$ |u(t)|_{Lp_c}≥ |log(T-t)|^{CN,p} { as} \ \ t→ T.