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

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