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(R) x Hs-1(R) with s > 3/2 was obtained. Under some assumptions, the uniqueness and existence for the weak solutions in lower Sobolev space Hs(R) x Hs-1(R) with 0 < s <= 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.