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Blow-up phemomenon for the Geng-Xue system and related models

Song Liu, Zhaoyang Yin

TL;DR

This work analyzes the Geng-Xue system with cubic nonlinearity, establishing both a blow-up criterion in Besov and Sobolev spaces and a finite-time blow-up mechanism for data satisfying a pointwise gradient condition, then extending the approach to the two-component $b$-family with cubic nonlinearity. The key methods combine Littlewood–Paley/Besov space techniques with characteristic-based arguments to control nonlocal terms and derive Riccati-type inequalities that force singularity formation. The results advance understanding of finite-time blow-up and ill-posedness in CH-type systems, including nonlocal convolution structures and parameter-dependent extensions. Overall, the paper provides rigorous blow-up thresholds and constructive blow-up scenarios, contributing to the theory of nonlinear dispersive PDEs with peakon-type dynamics and nonlocal nonlinearities.

Abstract

In this paper, we consider the Cauchy problem of the Geng-Xue system with cubic nonlinearity. Firstly, we prove a blow-up criteria in the low besov space. Secondly, we prove the blow-up phenomenon by using the method which does not require any conservation law. Finally, we extend our results to the b-family of two-component system with cubic nonlinearity.

Blow-up phemomenon for the Geng-Xue system and related models

TL;DR

This work analyzes the Geng-Xue system with cubic nonlinearity, establishing both a blow-up criterion in Besov and Sobolev spaces and a finite-time blow-up mechanism for data satisfying a pointwise gradient condition, then extending the approach to the two-component -family with cubic nonlinearity. The key methods combine Littlewood–Paley/Besov space techniques with characteristic-based arguments to control nonlocal terms and derive Riccati-type inequalities that force singularity formation. The results advance understanding of finite-time blow-up and ill-posedness in CH-type systems, including nonlocal convolution structures and parameter-dependent extensions. Overall, the paper provides rigorous blow-up thresholds and constructive blow-up scenarios, contributing to the theory of nonlinear dispersive PDEs with peakon-type dynamics and nonlocal nonlinearities.

Abstract

In this paper, we consider the Cauchy problem of the Geng-Xue system with cubic nonlinearity. Firstly, we prove a blow-up criteria in the low besov space. Secondly, we prove the blow-up phenomenon by using the method which does not require any conservation law. Finally, we extend our results to the b-family of two-component system with cubic nonlinearity.
Paper Structure (7 sections, 13 theorems, 144 equations, 1 table)

This paper contains 7 sections, 13 theorems, 144 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Application to the two-component b-family system with cubic nonlinearity
  4. Preliminaries
  5. Blow-up criteria
  6. Blow-up
  7. Application

Key Result

Theorem 1.1

$\text{(Blow-up criteria of the Geng-Xue system)}$ Let $(u_0,v_0) \in B^2_{2,1}(\mathbb{R}) \times B^2_{2,1}(\mathbb{R})$ and $T^*$ be the maximal existence time of the solution $(u,v)$ to the system (1.3). If $T < \infty$, then

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 17 more