Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation

TL;DR

This paper investigates the local well-posedness, blow-up phenomena, and peakon solutions of a generalized Camassa-Holm equation, providing criteria for blow-up and explicit blow-up rates using advanced mathematical tools.

Contribution

It establishes local well-posedness in Besov spaces, derives a blow-up criterion, and characterizes the blow-up rate and peakon solutions for the generalized Camassa-Holm equation.

Findings

Established local well-posedness in Besov spaces.

Derived a blow-up criterion for solutions.

Identified the exact blow-up rate and peakon solutions.

Abstract

In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a simply blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.Finally, we verify that the system possesses peakon solutions.