Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an interval

Abstract: In this paper we consider the initial boundary value problem (IBVP) for the nonlinear biharmonic Schr\"odinger equation posed on a bounded interval $(0,L)$ with non-homogeneous Navier or Dirichlet boundary conditions, respectively. For Navier boundary IBVP, we set up its local well-posedness if the initial data lies in $H^s(0, L)$ with $s\geq 0$ and $s\neq n+1/2, n\in \mathbb{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H_{loc}^{(s+3-j)/4}(\mathbb {R}^+)$, for $j=0,2$. For Dirichlet boundary IBVP the corresponding local well-posedness is obtained when $s>10/7$ and $s\neq n+1/2, n\in \mathbb{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H_{loc}^{(s+3-j)/4}(\mathbb {R}^+)$, for $j=0,1$.