Limit behavior of blow-up solutions for critical nonlinear Schrödinger equation with harmonic potential

Abstract

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a harmonic potential $i\phi_t+\frac{1}{2}\bigtriangleup\phi- \frac{1}{2}\omega^2|x|^2\phi+|\phi|^{4/N}\phi=0,\quad x \in R^N,\quad t \geq 0,$ which models the Bose-Einstein condensate. We establish the lower bound of blow-up rate as $t\rightarrow T$. Furthermore, the $L^2-$concentration property of the radially symmetric blow-up solutions is obtained.