Non-homogeneous initial boundary value problems for the biharmonic Schr\"odinger equation on an interval

TL;DR

This paper establishes local well-posedness results for the nonlinear biharmonic Schrödinger equation on a bounded interval with non-homogeneous boundary conditions, detailing regularity requirements for initial and boundary data.

Contribution

It provides the first well-posedness results for this equation with non-homogeneous boundary conditions, specifying optimal regularity spaces for initial and boundary data.

Findings

Well-posedness for Navier boundary conditions with initial data in H^s, s≥0

Well-posedness for Dirichlet boundary conditions with initial data in H^s, s>10/7

Optimal regularity spaces for boundary data are identified

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 $j=0,1$.