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This paper is concerned with a class of nonlocal dispersive models -- the$\theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation,{\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$(1-\partial_x^2)u_t+(1-\theta\partial_x^2)(\frac{u^2}{2})_x=(1-4\theta)(\frac{u_x^2}{2})_x, $$ including integrable equations such as theCamassa-Holm equation, $\theta=1/3$, and the Degasperis-Procesi equation,$\theta=1/4$, as special models. We investigate both global regularity ofsolutions and wave breaking phenomena for $\theta \in \mathbb{R}$. It is shownthat as $\theta$ increases regularity of solutions improves: (i) $0 <\theta <1/4$, the solution will blow up when the momentum of initial data satisfiescertain sign conditions; (ii) $1/4 \leq \theta < 1/2$, the solution will blowup when the slope of initial data is negative at one point; (iii) ${1/2} \leq\theta \leq 1$ and $\theta=\frac{2n}{2n-1}, n\in \mathbb{N}$, global existenceof strong solutions is ensured. Moreover, if the momentum of initial data has adefinite sign, then for any $\theta\in \mathbb{R}$ global smoothness of thecorresponding solution is proved. Proofs are either based on the use of someglobal invariants or based on exploration of favorable sign conditions ofquantities involving solution derivatives. Existence and uniqueness results ofglobal weak solutions for any $\theta \in \mathbb{R}$ are also presented. Forsome restricted range of parameters results here are equivalent to those knownfor the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-upphenomena, and global solutions for the b-equation, {\it J. reine angew.Math.}, {\bf 624} (2008)51--80.]

Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations