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

TL;DR

This paper studies the blow-up behavior and uniqueness of solutions to generalized Camassa-Holm-Kadomtsev-Petviashvili equations, providing criteria for blow-up and extending results to polynomial nonlinearities.

Contribution

It introduces a blow-up criterion independent of initial data regularity and extends blow-up results to polynomial nonlinearities, along with a Liouville-type uniqueness theorem.

Findings

Established a blow-up criterion independent of initial data regularity.

Proved blow-up and weighted blow-up results using characteristic lines and Riccati inequality.

Extended blow-up results to polynomially controlled nonlinearities, including classical cases.

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)=\kappa u+3u^2 $ for the classical CH-KP equations. Furthermore, a Liouville-type uniqueness theorem is established under the condition $g(u) \geq \gamma u^2$ with $u \neq 0$, $g(u)>\gamma u^2$.