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

We consider the nonlinear Schr\"odinger equation iuₜ=- u-|u|^p-1u in dimension N 3 in the L² super critical range 1+4N< p<N+2N-2. The corresponding scaling invariant space is Ḣ^sc 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=N2 (p-1) so that Ḣ^sc L^pc. Let u (t) Ḣ^sc Ḣ¹ be a radially symmetric blow up solution which blows up at 0<T<+, we prove that the scaling invariant L^pc norm also blows up with a lower bound: |u (t) |₋^₏₂ | (T-t) |^C₍, ₏ as \ \ t T.