1. Scattering and blowup problems for a class of nonlinear Schrödinger equations
- Author
-
Akahori, Takafumi, Kikuchi, Hiroaki, and Nawa, Hayato
- Subjects
Condensed Matter::Quantum Gases ,35Q55 ,Applied Mathematics ,35J20 ,35J61 ,Analysis - Abstract
We study the scattering and blowup problem for a class of nonlinear Schrödinger equations with general nonlinearities in the spirit of Kenig and Merle [17]. Our conditions on the nonlinearities allow us to treat a wider class of those than ever treated by several authors, so that we can prove the existence of a ground state (a standing-wave solution of minimal action) for any frequency $\omega > 0$. Once we get a ground state, a so-called potential-well scenario works well: for the nonlinear dynamics determined by the nonlinear Schrödinger equations, we define two invariant regions $A_{\omega, +}$ and $A_{\omega,-}$ for each $\omega > 0$ in $H^1(\mathbb{R}^d)$ such that any solution starting from $A_{\omega,+}$ behaves asymptotically free as $t\to\pm\infty$, one from $A_{\omega, -}$ blows up or grows up, and the ground state belongs to $\overline{A_{\omega, +}}\bigcap \overline{A_{\omega,-}}$. Our weaker assumptions as to the nonlinearities demand that we argue in a subtle way in proving the crucial properties of the solutions in the invariant regions.
- Published
- 2012