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Blow-Up Theory and Liouville-Type Theorem for Solutions of a Class of Generalized Camassa-Holm-Kadomtsev-Petviashvili Equations

Xueli Ke, Jiamin Wang, Aibin Zang

Abstract

We investigate the blow-up behavior and Liouville-type theorems of solutions to a class of generalized Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) equations with a generally smooth nonlinear term $g(u)$. First, using the continuation method, we establish a blow-up criterion that is independent of the regularity index of initial data. Under the assumption that $ g'(u)$ is uniformly bounded, we prove the blow-up theorem and a weighted blow-up result by means of characteristic lines, a priori estimates and the Riccati inequality. Moreover, we extend these blow-up results to the setting where $g'(u)$ is polynomially controlled, which includes typical nonlinearities such as $ g(u)=κu+3u^2 $ for the classical CH-KP equations. Furthermore, a Liouville-type uniqueness theorem is established under the condition $g(u) \geq γu^2$ with $u \neq 0$, $g(u)>γu^2$.

Blow-Up Theory and Liouville-Type Theorem for Solutions of a Class of Generalized Camassa-Holm-Kadomtsev-Petviashvili Equations

Abstract

We investigate the blow-up behavior and Liouville-type theorems of solutions to a class of generalized Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) equations with a generally smooth nonlinear term . First, using the continuation method, we establish a blow-up criterion that is independent of the regularity index of initial data. Under the assumption that is uniformly bounded, we prove the blow-up theorem and a weighted blow-up result by means of characteristic lines, a priori estimates and the Riccati inequality. Moreover, we extend these blow-up results to the setting where is polynomially controlled, which includes typical nonlinearities such as for the classical CH-KP equations. Furthermore, a Liouville-type uniqueness theorem is established under the condition with , .
Paper Structure (7 sections, 10 theorems, 152 equations)

This paper contains 7 sections, 10 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminary Lemmas
  3. Blow-up necessary condition of solution
  4. Blow-up sufficient conditions of solution
  5. The case of $g'\in L^\infty$
  6. The case of $g$ satisfies \ref{['cofg']}
  7. Liouville-type Theorem

Key Result

Theorem 1.1

Let $s \geq 2$ and $u \in C([0,T_{u_0}^*);X^s(\mathbb{R}^2)) \cap C^1([0,T_{u_0}^*);X^{s-2}(\mathbb{R}^2))$ be the unique solution to the system of equations eq:3 with initial condition $u_0 \in X^s(\mathbb{R}^2)$. If the maximal existence time $T_{u_0}^* < \infty$, then

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 7 more