The blow-up dynamics for the divergence Schr\"odinger equations with inhomogeneous nonlinearity

TL;DR

This paper investigates blow-up solutions for divergence Schrödinger equations with inhomogeneous nonlinearity, establishing upper bounds, concentration phenomena, and existence results for both radial and non-radial cases, extending classical NLS results.

Contribution

It introduces new blow-up rate bounds, concentration results, and existence proofs for solutions of inhomogeneous divergence Schrödinger equations, generalizing classical NLS findings.

Findings

Upper bound on blow-up rate for radial solutions

$L^2$-norm concentration in mass-critical case

Existence of finite time blow-up solutions in non-radial case

Abstract

This paper is dedicated to the blow-up solution for the divergence Schr\"{o}dinger 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 $\dot{H}^{s_c}\cap \dot{W}^{1,2}_b$, 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 Rapha\"{e}l [Amer. J. Math. 130(4) (2008), pp. 945-978] for the classical NLS equations to the dINLS setting.