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) R d × T n , d ≥ 5, n = 1, 2, 3. The tori 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, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader [46,47,48] and the 4NLS analogue of Tzvetkov-Visciglia [58].

On scattering for generalized NLS on waveguide manifolds

“…There are four steps and we will discuss them step by step. The strategy has similar spirit with [43,44]…”

“…

In this paper, we prove the large data scattering for fractional nonlinear Schrödinger equations (FNLS) on waveguide manifolds R d × T, d ≥ 3. This result can be regarded as the fractional analogue of [43,44] and the waveguide analogue of [16]. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.

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On the decay property of the cubic fourth-order Schrödinger equation

“…For dispersive models rather than 4NLS on Euclidean spaces (with scattering behavior), similar method may be applied to obtain the nonlinear decay property (i.e. the nonlinear solutions enjoy the same pointwise decay property as the linear solutions), such as, higher order (more than four) NLS, 4NLS on waveguide manifolds (see [31] for a recent result), NLS on waveguide manifolds (see [11,15,32] for example), NLS with partial harmonic potentials (see [1,3,12]), resonant system (see [4,30]), nonlinear wave equations (see [29]), Klein-Gordon equation (see [29]). We did not list them explicitly.…”

“…(See Remark A.3. in [31].) One can show the decay property in the sense of u(t, x) L r x → 0 as t → ∞ for some r > 2 which is sufficient to show the scattering for a subcritical model.…”

On scattering asymptotics for the 2D cubic resonant system

“…One may consider other related problems such as 'other dispersive equations on waveguides' and 'NLS on other manifolds/product spaces'. See [16,31,38] for examples (Klein-Gordon equations on waveguides, Fractional NLS on waveguides and Fourth order NLS on waveguides respectively). 7.2.…”