November/December 2012 Scattering and blowup problems for a class of nonlinear Schrödinger equations
Takafumi Akahori, Hiroaki Kikuchi, Hayato Nawa
Differential Integral Equations 25(11/12): 1075-1118 (November/December 2012). DOI: 10.57262/die/1356012252


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.


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Takafumi Akahori. Hiroaki Kikuchi. Hayato Nawa. "Scattering and blowup problems for a class of nonlinear Schrödinger equations." Differential Integral Equations 25 (11/12) 1075 - 1118, November/December 2012.


Published: November/December 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1274.35353
MathSciNet: MR3013405
Digital Object Identifier: 10.57262/die/1356012252

Primary: 35J20 , 35J61 , 35Q55

Rights: Copyright © 2012 Khayyam Publishing, Inc.


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Vol.25 • No. 11/12 • November/December 2012
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