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

TL;DR

This paper proves that for certain radial supercritical nonlinear Schrödinger equations, the critical norm must blow up at a logarithmic rate as solutions approach finite-time blow-up.

Contribution

It establishes the blow-up of the scaling invariant $L^{p_c}$ norm with a logarithmic lower bound for radial solutions in the supercritical regime.

Findings

The $L^{p_c}$ norm diverges as $t$ approaches blow-up time.

Blow-up rate of the norm is at least logarithmic.

Results apply to physically relevant case $N=p=3$.

Abstract

We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ 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}}\geq |\log(T-t)|^{C_{N,p}} $ as $t\to T$.