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 × Tⁿ, d ≥ 5, n = 1, 2, 3. The torus component Tⁿ can be generalized to n-dimensional compact manifolds Mⁿ. 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].