Annals of Mathematical Sciences and Applications

Volume 9 (2024)

Number 1

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

Pages: 3 – 55

DOI: https://dx.doi.org/10.4310/AMSA.2024.v9.n1.a1

Authors

Junfeng Li (School of Mathematical Sciences, Dalian University of Technology, Dalian, L.N., China)

Chuang Zheng (School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Abstract

In this paper we consider the initial boundary value problem (IBVP) for the nonlinear biharmonic Schrödinger 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 $s \neq n + 1/2, n \in \mathcal{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H^{(s+3-j)/4}_{loc} (\mathcal{R}^+)$, for $j = 0, 2$. For Dirichlet boundary IBVP the corresponding local well-posedness is obtained when $s \gt 10/7$ and $s \neq n+1/2, n \in \mathcal{N}$, and the boundary data are selected from the appropriate spaces with optimal regularities, i.e., the $j$-th order data are chosen in $H^{(s+3-j)/4}_{loc} (\mathcal{R}^+)$, for $j = 0, 1$.

Keywords

biharmonic Schrödinger equation, initial boundary value problems, boundary integral method, Navier boundary condition, Dirichlet boundary condition

2010 Mathematics Subject Classification

Primary 35B30, 35Q40. Secondary 31B30, 35Q55.

Received 30 July 2022

Accepted 6 October 2023

Published 5 April 2024