Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally wellposed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. In addition, the persistence properties of the strong solutions for the equation are obtained.

Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from nonstretching invariant curve flows in centro-equiaffine geometries, pseudospherical surfaces and affine surfaces are given.

Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equationA new Camassa-Holm system with two-component admitting peakon solitons is proposed. The local well posedness for the system is established. A criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained, and an existence result for a class of local weak solution is also given.

Considered herein is a modified two-component periodic Camassa-Holm system with peakons. The local well-posedness and low regularity result of solutions are established. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail and the exact blow-up rate is also obtained.

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