# American Institute of Mathematical Sciences

September  2017, 37(9): 5037-5048. doi: 10.3934/dcds.2017217

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

 Mathematics Department, Princeton University, Princeton, New Jersey, 08544, USA

Received  November 2016 Revised  April 2017 Published  June 2017

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.

Citation: Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217
##### References:
 [1] T. Cazenave and F. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529. Google Scholar [2] 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. Google Scholar [3] J. Ginibre and N. Hayashi, Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions, Math. Z., 219 (1995), 119-140. doi: 10.1007/BF02572354. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] Y. Kawahara, Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in $3D$, Differential Integral Equations, 18 (2005), 169-194. Google Scholar [8] 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. Google Scholar [9] W. Strauss, Nonlinear scattering theory at low energy, J. Fun. Anal., 41 (1981), 110-133. doi: 10.1016/0022-1236(81)90063-X. Google Scholar

show all references

##### References:
 [1] T. Cazenave and F. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529. Google Scholar [2] 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. Google Scholar [3] J. Ginibre and N. Hayashi, Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions, Math. Z., 219 (1995), 119-140. doi: 10.1007/BF02572354. Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] Y. Kawahara, Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in $3D$, Differential Integral Equations, 18 (2005), 169-194. Google Scholar [8] 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. Google Scholar [9] W. Strauss, Nonlinear scattering theory at low energy, J. Fun. Anal., 41 (1981), 110-133. doi: 10.1016/0022-1236(81)90063-X. Google Scholar
 [1] Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009 [2] Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323 [3] Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707 [4] Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385 [5] Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 [6] Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212 [7] Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 [8] Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 [9] Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004 [10] Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033 [11] Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845 [12] Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 [13] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [14] Daiwen Huang, Jingjun Zhang. Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1753-1773. doi: 10.3934/dcdss.2016073 [15] Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028 [16] Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83 [17] Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93 [18] Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429 [19] Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383 [20] Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705

2018 Impact Factor: 1.143