The blow-up dynamics for the divergence Schrödinger equations with inhomogeneous nonlinearity

Abstract

This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where $2-n<b<2$, $c>b-2$, and $np-2c<(2-b)(p+2)$. First, for radial blow-up solutions in $W_b^{1,2}$, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an $L^2$-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo-Nirenberg inequalities, we show the existence of finite time blow-up solutions in ${\dot{H}}^{s_c}\cap {\dot{W}}_b^{1,2}$, where ${\dot{H}}^{s_c}=(-\Delta {)}^{-\frac{s_c}{2}}{L}^2$, and ${\dot{W}}_b^{1,2}=|x{|}^{-\frac{b}{2}}(-\Delta {)}^{-\frac{1}{2}}{L}^2$. As an application, we obtain a lower bound for this blow-up rate, generalizing the work of Merle and Raphaël [Amer. J. Math. 130(4) (2008), pp. 945-978] for the classical NLS equations to the dINLS setting.

47 pages