Well-posedness and wave breaking for a shallow water wave model with large amplitude

Abstract

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.