On the Cauchy problem for a wave-structure interaction problem.

In this paper, we considered a wave-structure interaction problem modeling surface water waves (governed by a Boussinesq system) interacting with a fixed partially immersed object and vertical lateral walls, which can be reduced to a transmission problem for a Boussinesq system. The local well-posedness of the strong solutions for the Cauchy problem in Sobolev space $ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ with $ s>\frac{3}{2} $ was obtained. Under some assumptions, the uniqueness and existence for the weak solutions in lower Sobolev space $ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ with $ 0<s\leq 3/2 $ is also established via a limiting procedure. Moreover, a precise blow up criterion (i.e., the solution remains bounded but only the slope of component $ q $ becomes unbounded in finite time) was determined. [ABSTRACT FROM AUTHOR]