Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) ${\mathbb{R}}^d\times {\mathbb{T}}^n$, $d\ge 5$, $n=1,2,3$. The tori component ${\mathbb{T}}^n$ can be generalized to $n$-dimensional compact manifolds ${\mathcal{M}}^n$. First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader \cite{Pau2,Pau1,Pau3} and the 4NLS analogue of Tzvetkov-Visciglia \cite{TV2}.

25 pages. Comments are welcome!

References in corpus (3)

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• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces