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, utt−uxx+(u 2 )xx+uxxxx=0, x∈(0,1), t>0, u(x,0)= ϕ(x), ut(x,0)= ψ(x), u(0,t)=h1(t), u(1,t)=h2(t), uxx(0,t)=h3(t), uxx(1,t)=h4(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 ϕ, ψ lie in H s (0,1) and H s−2 (0,1), respectively, and the naturally com- patible boundary data h1, h2 in the space H (s+1)/2 loc (R + ), and h3, h4 in the the space of H (s−1)/2 loc (R + ) with optimal regularity.