TY - JOUR
T1 - GLOBAL WELL-POSEDNESS AND SCATTERING FOR FOURTH-ORDER SCHRÖDINGER EQUATIONS ON WAVEGUIDE MANIFOLDS
AU - Yu, Xueying
AU - Yue, Haitian
AU - Zhao, Zehua
N1 - Publisher Copyright:
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
PY - 2024/2
Y1 - 2024/2
N2 - 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].
AB - 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].
KW - fourth-order Schrödinger equation
KW - interaction Morawetz estimate
KW - scattering
KW - Strichartz estimate
KW - waveguide manifold
KW - well-posedness
UR - http://www.scopus.com/inward/record.url?scp=85195582472&partnerID=8YFLogxK
U2 - 10.1137/22M1529312
DO - 10.1137/22M1529312
M3 - Article
AN - SCOPUS:85195582472
SN - 0036-1410
VL - 56
SP - 1427
EP - 1458
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
IS - 1
ER -