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Existence and uniqueness of the global conservative solutions for the generalized Camassa-Holm equation with dual-power nonlinearities

Xiaoxin Chen, Jian Chen, Zhaoyang Yin

Abstract

In this paper, we investigate the global conservative solutions to the generalized Camassa-Holm equation with dual-power nonlinearities. By introducing a new set of variables, we transform the original equation into an equivalent semi-linear system, which allows us to establish the global existence of conservative solutions. Furthermore, for a given global conservative solution, we construct some auxiliary variables tailored to its specific structure and demonstrate that they satisfy a semi-linear system with a unique solution, thereby deriving the uniqueness of conservative solutions to the original equation.

Existence and uniqueness of the global conservative solutions for the generalized Camassa-Holm equation with dual-power nonlinearities

Abstract

In this paper, we investigate the global conservative solutions to the generalized Camassa-Holm equation with dual-power nonlinearities. By introducing a new set of variables, we transform the original equation into an equivalent semi-linear system, which allows us to establish the global existence of conservative solutions. Furthermore, for a given global conservative solution, we construct some auxiliary variables tailored to its specific structure and demonstrate that they satisfy a semi-linear system with a unique solution, thereby deriving the uniqueness of conservative solutions to the original equation.
Paper Structure (9 sections, 11 theorems, 159 equations)

This paper contains 9 sections, 11 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and results
  3. Existence
  4. An equivalent semi-linear system
  5. Global solutions of the semi-linear system
  6. Global conservative solutions for the original equation
  7. Uniqueness
  8. Preliminary lemmas
  9. Proof of uniqueness

Key Result

Theorem 2.3

Let $s\neq0$, then for any initial data $u_0\in H^1(\mathbb{R})$, the Cauchy problem eq1 has a unique global conservative solution in the sense of Definition conservative.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 11 more