Singular solutions of the subcritical nonlinear Schrodinger equation

TL;DR

This paper investigates the existence of smooth solutions to the subcritical nonlinear Schrödinger equation that develop singularities in certain Lebesgue spaces, revealing conditions under which collapse occurs.

Contribution

It demonstrates the existence of smooth solutions that become singular in specific Lebesgue spaces for the subcritical nonlinear Schrödinger equation, extending understanding of solution behavior.

Findings

Solutions can collapse in $L^p$ for $p^*<p \,\le\, \infty$

Collapse occurs at any $p$ in the range $2<p\le\infty$

Singular solutions exist for the critical exponent $p=2\sigma+2$

Abstract

We show that the subcritical $d$-dimensional nonlinear Schr\"odinger equation $i \psi_t + \Delta \psi + |\psi|^{2 \sigma} \psi = 0$, where $1<\sigma d<2$, admits smooth solutions that become singular in~$L^p$ for $p^*<p \le \infty$, where $p^*:=\frac{\sigma d}{\sigma d -1}$. Since $\lim_{\sigma d \to 2-} p^* = 2$, these solutions can collapse at any $2<p \le \infty$, and in particular for $p = 2 \sigma+2$.