Peakon solutions and analytical properties for the Camassa-Holm type equations with quadratic nonlinearities

TL;DR

This paper investigates the mathematical properties of a class of Camassa-Holm-type equations with quadratic nonlinearities, focusing on peakon solutions, well-posedness, blow-up criteria, and ill-posedness in specific function spaces.

Contribution

It derives the multi-peakon dynamical system and analyzes the analytical properties, including well-posedness, blow-up conditions, and ill-posedness in Besov spaces.

Findings

Established local well-posedness in Besov spaces.

Provided criteria for finite-time blow-up.

Proved ill-posedness in $B_{2, Infty}^{3/2}$ space.

Abstract

In this paper, we derive the multi-peakon dynamical system of a class of Camassa-Holm-type equations with quadratic nonlinearities. We also consider the analytical properties for the Cauchy problem. Firstly, we establish local well-posedness of solutions in Besov spaces and then provide the blow-up criteria. Subsequently, we impose appropriate sufficient conditions on the initial data to guaranty that the corresponding solution either exists globally or blows up in a finite time. Finally, we prove the ill-posedness in the Besov space $B_{2,\infty}^{3/2}$ by utilizing the non-traveling wave solutions.