A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain, {u(tt) - u(xx) + (u(2))(xx) + u(xxxx) = 0, x is an element of(0,1), t > 0, u(x,0) = phi(x), u(t)(x,0) = psi(x), u(0,t) = h(1)(t), u(1,t) = h(2)(t), u(xx)(0,t) = h(3)(t), u(xx)(1,t) = h(4)(t). It is shown that the IBVP is locally well-posed in the space H-s(0,1) for any s >= 0 with the initial data phi, psi lie in Hs(0,1) and Hs-2(0,1), respectively, and the naturally compatible boundary data h(1), h(2) in the space H-loc((s + 1)/2)(R+), and h(3), h(4) in the the space of H-loc((s-1)/2)(R+) with optimal regularity.