On the Unboundedness of Higher Regularity Sobolev Norms of Solutions for the Critical Schr\"odinger-Debye System with Vanishing Relaxation Delay

TL;DR

This paper investigates the Schr"odinger-Debye system in 3 and 4 dimensions, establishing global well-posedness in 3D and analyzing blow-up and unboundedness of higher regularity norms in 4D, especially as the delay parameter approaches zero.

Contribution

It extends well-posedness results for the Schr"odinger-Debye system and demonstrates unboundedness of higher Sobolev norms in the energy-critical 4D case, highlighting effects of vanishing delay.

Findings

Global well-posedness in 3D for broad initial data

Blow-up or unbounded higher norms in 4D for negative energy data

Global existence for small initial data in 4D

Abstract

We consider the Schr\"odinger-Debye system in $\mathbb{R}^n$, for $n=3,4$. Developing on previously known local well-posedness results, we start by establishing global well-posedness in $H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ for a broad class of initial data. We then concentrate on the initial value problem in $n=4$, which is the energy-critical dimension for the corresponding cubic nonlinear Schr\"odinger equation. We start by proving local well-posedness in $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$. Then, for the focusing case of the system, we derive a virial type identity and use it to prove that for radially symmetric smooth initial data with negative energy, there is a positive time $T_0$, depending only on the data, for which, either the $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ solutions blow-up in $[0,T_0]$, or the higher regularity Sobolev norms are unbounded on the intervals $[0, T]$, for $T>T_0$, as the delay parameter vanishes. We finish by presenting a global well-posedness result for regular initial data which is small in the $H^1(\mathbb{R}^4)\times H^1(\mathbb{R}^4)$ norm.