Convergence to scattering states in the nonlinear Schrödinger equation (Q2764515)

The author studies global solutions of the nonlinear Schrödinger equation NEWLINE\[NEWLINEiu_t+\Delta u+\lambda| u| ^\alpha u =0,NEWLINE\]NEWLINE with \(\lambda\in \mathbb{R}\) and \(\alpha\in (0,\frac{4}{N-2})(\alpha\in (2,\infty)\;\text{ if}\;N=1)\) in the Hilbert space \(X = H^1(\mathbb{R}^N)\cap L^2(| x| ^2;dx) \). If \(T(-t)u(t)-u_{\pm}\to 0 \) in \(X\) for \(t\rightarrow \pm\infty\), then \(u_{\pm}\) are called scattering states. The main goal is to study the asymptotic behavior of \(\| u(t)-T(t)u_{\pm}\| _X\) for \(t\rightarrow\pm \infty\) if \(u_\pm\) are such scattering states. For \(\lambda \neq 0\) and under some assumptions for \(\alpha\) and \(N\) it is shown that this norm tends to zero. For example, this holds if \(3\leq N \leq 5\) and \(\frac{4}{N-2}>\alpha > \frac{8}{N+2}\). For some other values of \(\alpha\) and \(N\) it is proved that NEWLINE\[NEWLINE \sup_{t\geq 0}\| u(t)-T(t)u_+\| _X < \infty,\qquad \sup_{t\leq 0} \| u(t) - T(t)u_-\| _X < \infty. NEWLINE\]NEWLINE Finally, for \(\lambda > 0\) and \(\frac{2}{N} < \alpha < \frac{4}{N-2}\) the norm difference \(| \;\| u(t)\| _X-\| T(t)u_{\pm}\| _X| \) is estimated.