Considered herein is the Cauchy problem of a model for shallow water waves of large amplitude. Using Littlewood–Paley decomposition and transport equation theory, we establish the local well-posedness of the equation in Besov spaces \(B^s_{p,r} \) with \(1\le p,r \le +\infty \) and \(s>\max \{1+\frac{1}{p },\frac{3}{2}\}\) (and also in Sobolev spaces \(H^s=B^s_{2,2}\) with \(s>3/2\)). Then, the precise blow-up mechanism for the strong solutions is determined in the lowest Sobolev space \(H^s \) with \(s>3/2\). Our results improve the corresponding work for this model in Quirchmayr (J Evol Equ 16:539–567, 2016), in which the Sobolev index \(s=3\) is required. In addition, we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted \(L_{p,\phi }:=L_p(\mathbb {R},\phi ^pdx)\) spaces.

中文翻译：

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大振幅浅水波模型的适定性与破波

这里考虑的是大振幅浅水波模型的柯西问题。使用Littlewood–Paley分解和输运方程理论，我们在Besov空间\（B ^ s_ {p，r} \）中以\（1 \ le p，r \ le + \ infty \ ）和\（s> \ max \ {1+ \ frac {1} {p}，\ frac {3} {2} \} \）（以及Sobolev空间\（H ^ s = B ^ s_ {2 ，2} \）和\（s> 3/2 \））。然后，在具有（（s> 3/2 \））的最低Sobolev空间\（H ^ s \）中确定用于强解的精确爆炸机制。我们的结果改进了该模型在Quirchmayr（J Evol Equ 16：539–567，2016）中的相应工作，其中Sobolev指数\（s = 3 \）是必需的。此外，如果初始数据位于加权的\（L_ {p，\ phi}：= L_p（\ mathbb {R}，\ phi ^ pdx）\）空格。