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Considered in this paper is the modified Camassa-Holm equation with cubicnonlinearity, which is integrable and admits the single peaked solitons andmulti-peakon solutions. The short-wave limit of this equation is known as theshort-pulse equation. The main investigation is the Cauchy problem of themodified Camassa-Holm equation with qualitative properties of its solutions. Itis firstly shown that the equation is locally well-posed in a range of theBesov spaces. The blow-up scenario and the lower bound of the maximal time ofexistence are then determined. A blow-up mechanism for solutions with certaininitial profiles is described in detail and nonexistence of the smoothtraveling wave solutions is also demonstrated. In addition, the persistenceproperties of the strong solutions for the equation are obtained.

On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity