Local well-posedness, wave breaking and ill-posedness for the unidirectional non-local wave model

In this paper, we first establish local well-posedness (existence, uniqueness and continuous dependence) for the Cauchy problem associated with the unidirectional non-local wave model in the Besov spaces Bˢp,r with s>1+1/p and 1≤ p,r≤∞, as well as in the critical Besov spaces B1+1/pp,1 with 1≤ p<∞. Next, by exploiting the fine structure of the equation and performing a delicate analysis of the commutator terms, we establish a new sufficient condition for wave breaking without using any conservation law. Finally, we prove that the model is ill-posed in B1+1/pp,r for 1≤ p≤∞ and $1