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Zbl 1158.35416
Gan, Zaihui; Zhang, Jian
Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system. (English)
[J] Commun. Math. Phys. 283, No. 1, 93-125 (2008). ISSN 0010-3616; ISSN 1432-0916/e

Summary: This paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave $\varphi(t,x) = e^{i\omega t} u(x)$ for the system: $$i\varphi_{t}+\Delta \varphi+a|\varphi|^{p-1}\varphi+b E_{1}(|\varphi|^{2})\varphi=0,\quad t\geq 0,\quad x\in \Bbb R^N, \tag{DS}$$ 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 existance and blowup of the solution to the Cauchy problem for (DS) provided $1+\frac 4n\leq p\leq\frac{N+2}{(N-2)^+}$. 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 blowup for any $\omega >0$ by combining the former results.

MSC 2000:: *35Q53 KdV-like equations
35B40 Asymptotic behavior of solutions of PDE
35B35 Stability of solutions of PDE
35A15 Variational methods (PDE)

Keywords: Davey-Stewartson system; standing waves; instability; cross-invariant manifolds; sharp threshold

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