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

Abstract

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 $B_{p,r}^s$ with $s>1+\frac{1}{p}$ and $1\le p,r\le \infty$, as well as in the critical Besov spaces $B_{p,1}^{1+\frac{1}{p}}$ with $1\le p<\infty$. 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 $B_{p,r}^{1+\frac{1}{p}}$ for $1\le p\le \infty$ and $1<r\le \infty$. Our analysis further clarifies the well-posedness theory and wave-breaking behavior of the unidirectional non-local wave model arising from the motion of a cold plasma in a magnetic field, which refines and improves previously known related results.