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Blow-up phemomenon for the 3-component Degasperis-Procesi equation

Song Liu, Zhaoyang Yin

Abstract

In this paper, we consider the Cauchy problem of the 3-component Degasperis-Procesi equation. Firstly, we discuss a local well-posedness result and a blow-up criterion in the low besov space. Secondly, we study the blow-up phenomenon by using the method which does not require any conservation law. Finally, we investigate some persistence properties.

Blow-up phemomenon for the 3-component Degasperis-Procesi equation

Abstract

In this paper, we consider the Cauchy problem of the 3-component Degasperis-Procesi equation. Firstly, we discuss a local well-posedness result and a blow-up criterion in the low besov space. Secondly, we study the blow-up phenomenon by using the method which does not require any conservation law. Finally, we investigate some persistence properties.
Paper Structure (7 sections, 18 theorems, 160 equations)

This paper contains 7 sections, 18 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Local well-posedness
  5. Blow-up
  6. Persistence properties
  7. Declarations

Key Result

Theorem 1.1

$\text{(local well-posedness)}$ If $s>\frac{3}{2}$ and $(u_0,v_0,\rho_{0}-1) \in H^s\times H^s\times H^s$ on the line or the circle, then there exists $T>0$ and a unique $(u,v,\rho-1)\in C([0,T];(H^s)^3)$ of the system (1.3) satisfying the following size estimate and lifespan where $c_s >0$ is a constant depending on $s$. Furthermore, the data-to-solution map is continuous but not uniformly conti

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 2.1
  • Proposition 2.2
  • ...and 14 more