American Institute of Mathematical Sciences

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September  2007, 8(2): 357-367. doi: 10.3934/dcdsb.2007.8.357

Sharp global existence and blowing up results for inhomogeneous Schrödinger equations

Jianqing Chen 1, and  Boling Guo 2,

1. 

Department of Mathematics, Fujian Normal University, Fuzhou, 350007, China
2. 

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088

Received  September 2006 Revised  March 2007 Published  June 2007

In this paper, we first give an important interpolation inequality. Secondly, we use this inequality to prove the existence of local and global solutions of an inhomogeneous Schrödinger equation. Thirdly, we construct several invariant sets and prove the existence of blowing up solutions. Finally, we prove that for any $\omega>0$ the standing wave $e^{i \omega t} \phi (x)$ related to the ground state solution $\phi$ is strongly unstable.
Keywords: global existence, strong instability., blowing up, Interpolation inequality.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J2.
Citation: Jianqing Chen, Boling Guo. Sharp global existence and blowing up results for inhomogeneous Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 357-367. doi: 10.3934/dcdsb.2007.8.357
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