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Sharp Threshold of Global Existence and Instability of Standing Wave for a Davey-Stewartson System

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Abstract

This paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave \(\phi(t,x) = e^{i\omega t} u(x)\) for the system:

$$i\phi_{t}+\Delta \phi+a|\phi|^{p-1}\phi+b E_{1}(|\phi|^{2})\phi\,=\,0,\quad t\,\geq 0,\quad x\in {\bf R}^{N}, \quad \quad \quad ({\rm DS})$$

where \(a > 0, b > 0, 1 < p < \frac{N+2}{(N-2)^{+}}\) and \(N\,\in\,\{2,3\}\). Firstly, by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution to the Cauchy problem for (DS) provided \(1+\frac{4}{N}\,\leq p\, < \frac{N+2}{(N-2)^{+}}\) . Secondly, by using the scaling argument, we show how small the initial data are for the global solutions to exist. Finally, we prove the strong instability of the standing waves with finite time blow up for any ω > 0 by combining the former results.

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Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610068, P. R. China

    Zaihui Gan & Jian Zhang

  2. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Zaihui Gan

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  1. Zaihui Gan
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  2. Jian Zhang
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Correspondence to Zaihui Gan.

Additional information

Communicated by P. Constantin

This work is supported by Sichuan Youth Science and Technology Foundation(07ZQ026-009) and The Institute of Mathematical Sciences at The Chinese University of Hong Kong.

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Gan, Z., Zhang, J. Sharp Threshold of Global Existence and Instability of Standing Wave for a Davey-Stewartson System. Commun. Math. Phys. 283, 93–125 (2008). https://doi.org/10.1007/s00220-008-0456-y

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