Blow-up for the b-family of equations

TL;DR

This paper investigates the blow-up phenomena in the b-family of equations on the torus, providing criteria for finite-time singularity formation in models including Camassa-Holm and Degasperis-Procesi.

Contribution

It establishes a local-in-space blow-up criterion for the b-family of equations, extending understanding of singularity formation in these nonlinear wave models.

Findings

Blow-up can occur under specific initial conditions.

The criterion applies to various models within the b-family.

Finite-time singularities are characterized by certain initial data.

Abstract

In this paper we consider the $b$-family of equations on the torus $u\_t- u\_{txx}+ (b+1) u u\_x=b u\_x u\_{xx} + u u\_{xxx}$, which for appropriate values of $b$ reduces to well-known models, such as the Camassa-Holm equation or the Degasperis-Procesi equation. We establish a local-in-space blow-up criterion