Scattering theory for the fourth-order Schrödinger equation in low dimensions (Q2845633)

The authors consider the defocusing fourth order Schrödinger equation with power nonlinearity in space dimension \(1\leq d\leq 4\), which is energy subcritical. They also assume that the nonlinearity is mass supercritical. In this context, they prove a definitive result of scattering: any solution in the energy space behaves in the energy space as a solution of the free problem. Due to the lack of Morawetz type estimate for this equation, the authors have used the ``concentration-compactness-rigidity'' strategy of \textit{C. E. Kenig} and \textit{F. Merle} [Invent. Math. 166, No. 3, 645--675 (2006; Zbl 1115.35125)]. The rigidity part requires a new Virial estimate inspired by previous results in a similar context due to [\textit{B. Pausader}, Indiana Univ. Math. J. 59, No. 3, 791--822 (2010; Zbl 1214.35071); \textit{B. Pausader} and \textit{S. Shao}, J. Hyperbolic Differ. Equ. 7, No. 4, 651--705 (2010; Zbl 1232.35156)].