The Cauchy problem and wave-breaking phenomenon for a generalized sine-type FORQ/mCH equation

TL;DR

This paper investigates the initial value problem and wave-breaking behavior of a sine-type modified Camassa-Holm equation, establishing local well-posedness, blow-up criteria, and conditions for wave-breaking using advanced mathematical theories.

Contribution

It introduces a new analysis of the sine-FORQ/mCH equation, providing the first local well-posedness results and explicit blow-up conditions in Besov and Sobolev spaces.

Findings

Established local well-posedness in Besov spaces.

Derived blow-up criteria and identified blow-up quantities.

Provided conditions for wave-breaking based on initial data.

Abstract

In this paper, we are concerned with the Cauchy problem and wave-breaking phenomenon for a sine-type modified Camassa-Holm (alias sine-FORQ/mCH) equation. Employing the transport equations theory and the Littlewood-Paley theory, we first establish the local well-posedness for the strong solutions of the sine-FORQ/mCH equation in Besov spaces. In light of the Moser-type estimates, we are able to derive the blow-up criterion and the precise blow-up quantity of this equation in Sobolev spaces. We then give a sufficient condition with respect to the initial data to ensure the occurance of the wave-breaking phenomenon by trace the precise blow-up quantity along the characteristics associated with this equation.