(untitled) (Journal of Mathematical Study, 2022)
In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain, cases &u_tt-u_xx+(u^2)_xx+u_xxxx=0,x(0,1), t>0, &u(x,0)=φ(x), u_t(x,0)=ψ(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). cases 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 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.