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Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type

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  • We study a class of $3D$ quadratic Schrödinger equations as follows, $(\partial_t -i Δ) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah in [2], the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1× 1 \to 0$ type interaction (the size of input frequency is "1" and the size of output frequency is "0"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the $1× 0\to 1$ type interaction. The issue of strong $1× 1\to 0$ type interaction makes the global existence problem very delicate.

    In this paper, we show that, as long as there are "$ε$" derivatives inside the quadratic term $Q (u, \bar{u})$, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of $(\partial_t -i Δ)u = |u|^2 = u\bar{u}$, which was first proved by Ginibre-Hayashi [3]. Instead of using vector fields, we consider this problem purely in Fourier space.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q35.


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  •   T. Cazenave  and  F. Weissler , Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992) , 75-100.  doi: 10.1007/BF02099529.
      P. Germain , N. Masmoudi  and  J. Shatah , Global Solutions for $3D$ Quadratic Schrödinger Equations, Int. Math. Res. Notice, 2009 (2009) , 414-432.  doi: 10.1093/imrn/rnn135.
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      Z. Guo , L. Peng  and  B. Wang , Decay estimates for a class of wave equations, J. Func, Anal., 254 (2008) , 1642-1660.  doi: 10.1016/j.jfa.2007.12.010.
      N. Hayashi  and  P. Naumkin , On the quadratic nonlinear Schrödinger equation in three space dimensions, Int. Math. Res. Notice, 2000 (2000) , 115-132.  doi: 10.1155/S1073792800000088.
      M. Ikeda  and  T. Inui , Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015) , 571-581.  doi: 10.1007/s00028-015-0273-7.
      Y. Kawahara , Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in $3D$, Differential Integral Equations, 18 (2005) , 169-194. 
      S. Klainerman  and  G. Ponce , Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure. Appl. Math., 36 (1983) , 133-141.  doi: 10.1002/cpa.3160360106.
      W. Strauss , Nonlinear scattering theory at low energy, J. Fun. Anal., 41 (1981) , 110-133.  doi: 10.1016/0022-1236(81)90063-X.
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