Abstract
Fourth-order Schrödinger equations have been introduced by Karpman and Shagalov to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In this paper we investigate the cubic defocusing fourth-order Schrödinger equation i ∂ t u + Δ 2 u + | u | 2 u = 0 in arbitrary space dimension R n for arbitrary initial data. We prove that the equation is globally well-posed when n ⩽ 8 and ill-posed when n ⩾ 9 , with the additional important information that scattering holds true when 5 ⩽ n ⩽ 8 .
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