Global existence and propagation speed for a generalized Camassa-Holm model with both dissipation and dispersion

TL;DR

This paper analyzes a generalized Camassa-Holm model with dissipation and dispersion, establishing conditions for global solutions, blow-up scenarios, and propagation speed, unifying several integrable equations.

Contribution

It introduces a comprehensive analysis of a generalized Camassa-Holm model, including well-posedness, blow-up criteria, and speed propagation, encompassing multiple known equations as special cases.

Findings

Local well-posedness established

Conditions for global existence derived

Propagation speed for compactly supported data analyzed

Abstract

In this paper, we study a generalized Camassa-Holm (gCH) model with both dissipation and dispersion, which has (N + 1)-order nonlinearities and includes the following three integrable equations: the Camassa-Holm, the Degasperis-Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data is compactly supported.