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 $g(u)$. 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 $g^{\prime }(u)$ 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 $g^{\prime }(u)$ 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 $g(u)\ge \gamma u^2$ with $u\ne 0$, $g(u)>\gamma u^2$.

23 pages