Scattering theory for the defocusing fourth-order Schrödinger equation (Q2794866)

The authors consider the defocusing fourth-order nonlinear Schrödinger equation \(iu_t+\Delta^2u+|u|^pu=0\) posed in \(\mathbb R\times\mathbb R^d\) with \(d\geq 8\) and with the initial condition \(u(0,x)=u_0(x)\in \dot H_x^{s_c}(\mathbb R^d)\), \(s_c=\frac{d}{2}-\frac{p}{4}\). The purpose of the paper is to prove an existence result for a strong solution \(u\in C_t(K;\dot H_x^{s_c}(\mathbb R^d))\cap L_{t,x}^{\frac{d+4}{4}}(K\times \mathbb R^d)\) for every compact interval \(K\subset I\), that is, satisfying NEWLINE\[NEWLINEu(t,x)=e^{i(t-t_0)\Delta^2}u(t_0)-i\int_{t_0}^te^{i(t-t_0)\Delta^2}f(u(s))dsNEWLINE\]NEWLINE for every \(t,t_{0}\in I\). The main result of the paper proves that if \(s_c\geq 1\), if \(p\) is an even integer or \(s_c\in[1,2+p)\) otherwise and moreover \(s_c>2\), if \(d=8\), then a maximal life-span and strong solution which satisfies NEWLINE\[NEWLINE\left\| u\right\|_{L_t^\infty(I;\dot H_x^{s_c}\mathbb R^d))}<+\infty NEWLINE\]NEWLINE is global. Moreover, this global solution \(u\) scatters in the sense that NEWLINE\[NEWLINE\left\| u-e^{it\Delta^2}v_\pm\right\|_{\dot H_x^{s_c}(\mathbb R^d)}\rightarrow\inftyNEWLINE\]NEWLINE as \(t\rightarrow\pm\infty\) for some unique \(v_\pm\in\dot H_x^{s_c}(\mathbb R^d)\). For the proof, the authors first prove a local existence result and a refined Duhamel principle. They also use a refined dispersive estimate, Strichartz estimates and a nonlinear Bernstein inequality, and also concentration compactness tools.