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

Abstract

We consider the nonlinear Schrödinger equation $i{u}_t=-\Delta u-|u{|}^{p-1}u$ 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 $0<T<+\infty$, 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 $|u(t){|}_{L^{p_c}}\ge |\log (T-t){|}^{C_{N,p}}$ as $t\to T$.

This is a replacement of the pevious version which covered the case $1/2\le s_c<1$ only