The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

Türker Özsarı; Nermin Yolcu

doi:10.3934/cpaa.2019148

Article Contents

2019, Volume 18, Issue 6: 3285-3316. Doi: 10.3934/cpaa.2019148

This issue Previous Article Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity Next Article The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

Türker Özsarı^, and
Nermin Yolcu

Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, TURKEY

^* Corresponding author
^* Corresponding author

Received: July 31, 2018

Revised: January 31, 2019

Published: May 2019

Both authors are supported by TÜBİTAK 1001 Grant 117F449

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Abstract

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.

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Mathematics Subject Classification: Primary: 35Q55, 35C15, 35A01, 35A02, 35G31; Secondary: 35A22.

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Figure 1. The region $ D = D^+\cup D^- $

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Figure 3. Partitioning the boundary

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