Global Well-Posedness and Scattering for Fourth-Order Schrödinger Equations on Waveguide Manifolds

.In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) \(R^d× T^n\), \(d ≥ 5\), \(n=1,2,3\). The torus component \(T^n\) can be generalized to \(n\)-dimensional compact manifolds \(M^n\). 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 and the focusing scenario, and give some further remarks on this research line. This result can be regarded as the waveguide analogue of [C. Miao, G. Xu, and L. Zhao, J. Differential Equations, 251 (2011), pp. 3381–3402], [B. Pausader, Dyn. Partial Differ. Equ., 4 (2007), pp. 197–225], [B. Pausader, J. Funct. Anal., 256 (2009), pp. 2473–2517], [B. Pausader, Discrete Contin. Dyn. Syst., 24 (2009), pp. 1275–1292] and the 4NLS analogue of Tzvetkov and Visciglia [N. Tzvetkov and N. Visciglia, Rev. Mat. Iberoam., 32 (2016), pp. 1163–1188].