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) \ (Rᵈ 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. KeywordsStrichartz estimatefourth-order Schrödinger equationwaveguide manifoldwell-posednessscatteringinteraction Morawetz estimateMSC codes35Q5535R0137K0637L50