On small data scattering with cubic convolution nonlinearity (Q915002)

Let X be a Hilbert space, A a selfadjoint operator in X with dense domain, consider the problem \[ (1)\quad i\partial_ tu=Au+F(u),\quad t\in {\mathbb{R}},\quad \| u(t)-U_ 0(t)\phi_ -\|_ x\to 0\quad as\quad t\to -\infty, \] where \(\phi_ -\in X\) and \(U_ 0(t)=\exp (- iAt)\), \(t\in {\mathbb{R}}\). Write (1) in the integral form \((2)\quad u(t)=U_ 0(t)\phi_ -+\int^{t}_{-\infty}U_ 0(t-\tau)F(u(\tau))d\tau.\) Under some hypotheses there exists a \(\delta >0\) with: If \(\| \phi_ - \|_ x\leq \delta\), then there exists a unique solution of (2) such that \(\| u\| \leq \frac{4}{3}\| U_ 0(t)\phi_ -\|.\) Furthermore there exists a unique \(\phi_+\in X\) such that \(\| u(t)- U_ 0(t)\phi_+\|_ x\to 0\) as \(t\to \infty\). Thus, the scattering operaor \(S:\phi_ -\to \phi_+\) on a neighborhood of 0 in X is defined. The Schrödinger equation \((1/i)\partial_ tw=\Delta w+f(w),\) Klein- Gordon equation \(\partial_ t^ 2w=\Delta w-w+f(w)\) and the wave equation \(\partial_ t^ 2w=\Delta w+f(w)\) for \((x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}\) are considered, f(w) represents the nonlinearity \[ (f(w))(x)=((V*| w|^ 2)w)(x)=(\int V(x-y)| w(y)|^ 2dy)w(x) \] with \(| V(x)| \leq c| x|^{-\sigma}\) or \(V\in L^ z({\mathbb{R}}^ n)\) \((2\leq \sigma <n\), \(1\leq z\leq \frac{n}{2}\) Schrödinger and Klein-Gordon; \(4\leq \sigma <n\), \(1\leq z\leq \frac{n}{4}\) wave equation).