Differential and Integral Equations

Scattering and blowup problems for a class of nonlinear Schrödinger equations

Takafumi Akahori, Hiroaki Kikuchi, and Hayato Nawa

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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.

Article information

Source
Differential Integral Equations, Volume 25, Number 11/12 (2012), 1075-1118.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012252

Mathematical Reviews number (MathSciNet)
MR3013405

Zentralblatt MATH identifier
1274.35353

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J61: Semilinear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Citation

Akahori, Takafumi; Kikuchi, Hiroaki; Nawa, Hayato. Scattering and blowup problems for a class of nonlinear Schrödinger equations. Differential Integral Equations 25 (2012), no. 11/12, 1075--1118. https://projecteuclid.org/euclid.die/1356012252


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