Scattering theory for the defocusing fourth-order Schrödinger equation

Abstract

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) $i{u}_t+{\Delta }^{2}u+|u{|}^{p}u=0$ in dimension $d\ge 9$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^{\infty }(I;{\dot{H}}_x^{s_c}({\mathbb{R}}^d))$ with all $s_c:=\frac{d}{2}-\frac{4}{p}\ge 1$ if $p$ is an even integer or $s_c\in [1,2+p)$ otherwise, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS) and nonlinear wave equation (NLW). We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS, where we utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to exclude the existence of three scenarios: finite time blowup; soliton-like solution and low to high frequency cascade. Making use of the No-waste Duhamel formula, we deduce that the energy or mass of the finite time blow-up solution is zero and so get a contradiction. Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate and interpolation to kill the last two scenarios.

40pages. arXiv admin note: text overlap with arXiv:0812.2084 by other authors

References in corpus (4)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions $d\ge 9$
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Cited by in corpus (2)

• Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds
• On the decay property of the cubic fourth-order Schrödinger equation