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Article Contents

# A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$

• In this paper we study the threshold of global existence and blow-up for the solutions to the generalized 3D Davey-Stewartson equations \begin{equation*} \left\{ \begin{aligned} & iu_t + \Delta u + |u|^{p-1} u + E_1(|u|^2)u = 0, \quad t > 0, \ \ x\in \mathbb{R}^3, \\ & u(0,x) = u_0(x) \in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where $1 < p < \frac{7}{3}$ and the operator $E_1$ is given by $E_1(f) = \mathcal {F}^{-1} \left( \frac{\xi_1^2}{|\xi|^2} \mathcal{F}(f) \right)$. We construct two kinds of invariant sets under the evolution flow by analyzing the property of the upper bound function of the energy. Then we show that the solution exists globally for the initial function $u_0$ in first kind of the invariant sets, while the solution blows up in finite time for $u_0$ in another kind. We remark that the exponent $p$ is subcritical for the nonlinear Schrödinger equations for which blow-up solutions would not occur. The result shows that the occurrence of blow-up phenomenon is caused by the coupling mechanics of the Davey-Stewartson equations.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44.

 Citation:

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