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November  2019, 18(6): 3285-3316. doi: 10.3934/cpaa.2019148

## The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

 Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, TURKEY

* Corresponding author

Received  August 2018 Revised  February 2019 Published  May 2019

Fund Project: Both authors are supported by TÜBİTAK 1001 Grant 117F449

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.

Citation: Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
##### References:
 [1] B. Aksas and S.-E. Rebiai, Uniform stabilization of the fourth order Schrödinger equation, J. Math. Anal. Appl., 446 (2017), 1794-1813. doi: 10.1016/j.jmaa.2016.09.065. [2] K. Aoki, N. Hayashi and P. I. Naumkin, Global existence of small solutions for the fourth-order nonlinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 65, 18. doi: 10.1007/s00030-016-0420-z. [3] C. Audiard, Global Strichartz estimates for the Schrödinger equation with nonzero boundary conditions and applications, Ann. Inst. Fourier. [4] G. Baruch and G. Fibich, Singular solutions of the $L^2$-supercritical biharmonic nonlinear Schrödinger equation, Nonlinearity, 24 (2011), 1843-1859. doi: 10.1088/0951-7715/24/6/009. [5] A. Batal and T. Özsarı, Nonlinear Schrödinger equations on the half-line with nonlinear boundary conditions, Electron. J. Differential Equations, Paper No. 222, 20. [6] M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8. [7] J. L. Bona, S.-M. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for onedimensional nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 109 (2018), 1–66. doi: 10.1016/j.matpur.2017.11.001. [8] T. Boulenger and E. Lenzmann, Blowup for biharmonic NLS, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 503–544. doi: 10.24033/asens.2326. [9] Q. Bu, On well-posedness of the forced nonlinear Schrödinger equation, Appl. Anal., 46 (1992), 219-239. doi: 10.1080/00036819208840122. [10] R. Carroll and Q. Bu, Solution of the forced nonlinear Schrödinger (NLS) equation using PDE techniques, Appl. Anal., 41 (1991), 33-51. doi: 10.1080/00036819108840015. [11] S. Cui and C. Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(\Bbb R^n)$ and applications, Nonlinear Anal., 67 (2007), 687-707. doi: 10.1016/j.na.2006.06.020. [12] M. Dimakos and A. S. Fokas, The Poisson and the biharmonic equations in the interior of a convex polygon, Stud. Appl. Math., 134 (2015), 456-498. doi: 10.1111/sapm.12078. [13] V. D. Dinh, On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space, Dyn. Partial Differ. Equ., 14 (2017), 295-320. doi: 10.4310/DPDE.2017.v14.n3.a4. [14] V. D. Dinh, On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437. [15] V. D. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. [16] G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241. [17] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A, 453 (1997), 1411-1443. doi: 10.1098/rspa.1997.0077. [18] A. S. Fokas, A Unified Approach to Boundary Value Problems, vol. 78 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898717068. [19] A. S. Fokas, A. A. Himonas and D. Mantzavinos, The Korteweg–de Vries equation on the half-line, Nonlinearity, 29 (2016), 489-527. doi: 10.1088/0951-7715/29/2/489. [20] A. S. Fokas, A. A. Himonas and D. Mantzavinos, The nonlinear Schrödinger equation on the half-line, Trans. Amer. Math. Soc., 369 (2017), 681-709. doi: 10.1090/tran/6734. [21] C. Guo, Global existence of solutions for a fourth-order nonlinear Schrödinger equation in $n+1$ dimensions, Nonlinear Anal., 73 (2010), 555-563. doi: 10.1016/j.na.2010.03.052. [22] C. Guo, Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrödinger equations with combined power-type nonlinearities, J. Math. Anal. Appl., 392 (2012), 111-122. doi: 10.1016/j.jmaa.2012.03.028. [23] C. Hao, L. Hsiao and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265. doi: 10.1016/j.jmaa.2005.06.091. [24] C. Hao, L. Hsiao and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces, J. Math. Anal. Appl., 328 (2007), 58-83. doi: 10.1016/j.jmaa.2006.05.031. [25] N. Hayashi and P. I. Naumkin, Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377. doi: 10.1007/s00033-015-0524-z. [26] N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131. doi: 10.1016/j.na.2014.12.024. [27] N. Hayashi and P. I. Naumkin, Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905. doi: 10.1016/j.jde.2014.10.007. [28] N. Hayashi and P. I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 093502, 25. doi: 10.1063/1.4929657. [29] A. A. Himonas and D. Mantzavinos, The "good" Boussinesq equation on the half-line, J. Differential Equations, 258 (2015), 3107-3160. doi: 10.1016/j.jde.2015.01.005. [30] A. A. Himonas and D. Mantzavinos, Well-posedness of the nonlinear Schrödinger equation on the half-plane, arXiv: 1810.02395. [31] A. A. Himonas, D. Mantzavinos and F. Yan, Well-posedness of initial-boundary value problems for a reaction-diffusion equation, arXiv: 1810.05322. [32] J. Holmer, The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668. [33] V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrodinger-type equations, Phys. Rev. E, 53 (1996), R1336–R1339. doi: 10.1016/0375-9601(95)00752-0. [34] V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6. [35] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition doi: 10.1007/978-1-4939-2181-2. [36] C. Miao, H. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Nachr., 288 (2015), 798-823. doi: 10.1002/mana.201400012. [37] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749. doi: 10.1016/j.jde.2008.11.011. [38] B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1. [39] B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations, J. Differential Equations, 241 (2007), 237-278. doi: 10.1016/j.jde.2007.06.001. [40] B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009. [41] B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyperbolic Differ. Equ., 7 (2010), 651-705. doi: 10.1142/S0219891610002256. [42] B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191. doi: 10.1088/0951-7715/26/8/2175. [43] M. Ruzhansky, B. Wang and H. Zhang, Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl. (9), 105 (2016), 31–65. doi: 10.1016/j.matpur.2015.09.005. [44] J.-i. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568. doi: 10.1090/S0002-9939-04-07620-8. [45] J.-i. Segata, Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearity, Math. Methods Appl. Sci., 29 (2006), 1785-1800. doi: 10.1002/mma.751. [46] J.-i. Segata and A. Shimomura, Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity, J. Math. Kyoto Univ., 46 (2006), 439-456. doi: 10.1215/kjm/1250281786. [47] Y. Wang, Nonlinear fourth-order Schrödinger equations with radial data, Nonlinear Anal., 75 (2012), 2534-2541. doi: 10.1016/j.na.2011.10.047. [48] R. Wen and S. Chai, Well-posedness and exact controllability of a fourth order Schrödinger equation with variable coefficients and Neumann boundary control and collocated observation, Electron. J. Differential Equations, Paper No. 216, 17. [49] R. Wen, S. Chai and B.-Z. Guo, Well-posedness and exact controllability of fourth order Schrödinger equation with boundary control and collocated observation, SIAM J. Control Optim., 52 (2014), 365-396. doi: 10.1137/120902744. [50] R. Wen, S. Chai and B.-Z. Guo, Well-posedness and exact controllability of fourth-order Schrödinger equation with hinged boundary control and collocated observation, Math. Control Signals Systems, 28 (2016), Art. 22, 28. doi: 10.1007/s00498-016-0175-4. [51] J. Zhang and J. Zheng, Energy critical fourth-order Schrödinger equations with subcritical perturbations, Nonlinear Anal., 73 (2010), 1004-1014. doi: 10.1016/j.na.2010.04.027. [52] J. Zheng, Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity, Adv. Differential Equations, 16 (2011), 467-486. [53] S. Zhu, H. Yang and J. Zhang, Blow-up of rough solutions to the fourth-order nonlinear Schrödinger equation, Nonlinear Anal., 74 (2011), 6186-6201. doi: 10.1016/j.na.2011.05.096. [54] S. Zhu, J. Zhang and H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205. doi: 10.4310/DPDE.2010.v7.n2.a4. [55] S. Zhu, J. Zhang and H. Yang, Biharmonic nonlinear Schrödinger equation and the profile decomposition, Nonlinear Anal., 74 (2011), 6244-6255. doi: 10.1016/j.na.2011.06.004.

show all references

##### References:
 [1] B. Aksas and S.-E. Rebiai, Uniform stabilization of the fourth order Schrödinger equation, J. Math. Anal. Appl., 446 (2017), 1794-1813. doi: 10.1016/j.jmaa.2016.09.065. [2] K. Aoki, N. Hayashi and P. I. Naumkin, Global existence of small solutions for the fourth-order nonlinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 65, 18. doi: 10.1007/s00030-016-0420-z. [3] C. Audiard, Global Strichartz estimates for the Schrödinger equation with nonzero boundary conditions and applications, Ann. Inst. Fourier. [4] G. Baruch and G. Fibich, Singular solutions of the $L^2$-supercritical biharmonic nonlinear Schrödinger equation, Nonlinearity, 24 (2011), 1843-1859. doi: 10.1088/0951-7715/24/6/009. [5] A. Batal and T. Özsarı, Nonlinear Schrödinger equations on the half-line with nonlinear boundary conditions, Electron. J. Differential Equations, Paper No. 222, 20. [6] M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8. [7] J. L. Bona, S.-M. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for onedimensional nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 109 (2018), 1–66. doi: 10.1016/j.matpur.2017.11.001. [8] T. Boulenger and E. Lenzmann, Blowup for biharmonic NLS, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 503–544. doi: 10.24033/asens.2326. [9] Q. Bu, On well-posedness of the forced nonlinear Schrödinger equation, Appl. Anal., 46 (1992), 219-239. doi: 10.1080/00036819208840122. [10] R. Carroll and Q. Bu, Solution of the forced nonlinear Schrödinger (NLS) equation using PDE techniques, Appl. Anal., 41 (1991), 33-51. doi: 10.1080/00036819108840015. [11] S. Cui and C. Guo, Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces $H^s(\Bbb R^n)$ and applications, Nonlinear Anal., 67 (2007), 687-707. doi: 10.1016/j.na.2006.06.020. [12] M. Dimakos and A. S. Fokas, The Poisson and the biharmonic equations in the interior of a convex polygon, Stud. Appl. Math., 134 (2015), 456-498. doi: 10.1111/sapm.12078. [13] V. D. Dinh, On the focusing mass-critical nonlinear fourth-order Schrödinger equation below the energy space, Dyn. Partial Differ. Equ., 14 (2017), 295-320. doi: 10.4310/DPDE.2017.v14.n3.a4. [14] V. D. Dinh, On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437. [15] V. D. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. [16] G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241. [17] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A, 453 (1997), 1411-1443. doi: 10.1098/rspa.1997.0077. [18] A. S. Fokas, A Unified Approach to Boundary Value Problems, vol. 78 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898717068. [19] A. S. Fokas, A. A. Himonas and D. Mantzavinos, The Korteweg–de Vries equation on the half-line, Nonlinearity, 29 (2016), 489-527. doi: 10.1088/0951-7715/29/2/489. [20] A. S. Fokas, A. A. Himonas and D. Mantzavinos, The nonlinear Schrödinger equation on the half-line, Trans. Amer. Math. Soc., 369 (2017), 681-709. doi: 10.1090/tran/6734. [21] C. Guo, Global existence of solutions for a fourth-order nonlinear Schrödinger equation in $n+1$ dimensions, Nonlinear Anal., 73 (2010), 555-563. doi: 10.1016/j.na.2010.03.052. [22] C. Guo, Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrödinger equations with combined power-type nonlinearities, J. Math. Anal. Appl., 392 (2012), 111-122. doi: 10.1016/j.jmaa.2012.03.028. [23] C. Hao, L. Hsiao and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265. doi: 10.1016/j.jmaa.2005.06.091. [24] C. Hao, L. Hsiao and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces, J. Math. Anal. Appl., 328 (2007), 58-83. doi: 10.1016/j.jmaa.2006.05.031. [25] N. Hayashi and P. I. Naumkin, Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377. doi: 10.1007/s00033-015-0524-z. [26] N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131. doi: 10.1016/j.na.2014.12.024. [27] N. Hayashi and P. I. Naumkin, Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905. doi: 10.1016/j.jde.2014.10.007. [28] N. Hayashi and P. I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 093502, 25. doi: 10.1063/1.4929657. [29] A. A. Himonas and D. Mantzavinos, The "good" Boussinesq equation on the half-line, J. Differential Equations, 258 (2015), 3107-3160. doi: 10.1016/j.jde.2015.01.005. [30] A. A. Himonas and D. Mantzavinos, Well-posedness of the nonlinear Schrödinger equation on the half-plane, arXiv: 1810.02395. [31] A. A. Himonas, D. Mantzavinos and F. Yan, Well-posedness of initial-boundary value problems for a reaction-diffusion equation, arXiv: 1810.05322. [32] J. Holmer, The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668. [33] V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrodinger-type equations, Phys. Rev. E, 53 (1996), R1336–R1339. doi: 10.1016/0375-9601(95)00752-0. [34] V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6. [35] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition doi: 10.1007/978-1-4939-2181-2. [36] C. Miao, H. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Nachr., 288 (2015), 798-823. doi: 10.1002/mana.201400012. [37] C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749. doi: 10.1016/j.jde.2008.11.011. [38] B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1. [39] B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations, J. Differential Equations, 241 (2007), 237-278. doi: 10.1016/j.jde.2007.06.001. [40] B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009. [41] B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyperbolic Differ. Equ., 7 (2010), 651-705. doi: 10.1142/S0219891610002256. [42] B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191. doi: 10.1088/0951-7715/26/8/2175. [43] M. Ruzhansky, B. Wang and H. Zhang, Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl. (9), 105 (2016), 31–65. doi: 10.1016/j.matpur.2015.09.005. [44] J.-i. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568. doi: 10.1090/S0002-9939-04-07620-8. [45] J.-i. Segata, Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearity, Math. Methods Appl. Sci., 29 (2006), 1785-1800. doi: 10.1002/mma.751. [46] J.-i. Segata and A. Shimomura, Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity, J. Math. Kyoto Univ., 46 (2006), 439-456. doi: 10.1215/kjm/1250281786. [47] Y. Wang, Nonlinear fourth-order Schrödinger equations with radial data, Nonlinear Anal., 75 (2012), 2534-2541. doi: 10.1016/j.na.2011.10.047. [48] R. Wen and S. Chai, Well-posedness and exact controllability of a fourth order Schrödinger equation with variable coefficients and Neumann boundary control and collocated observation, Electron. J. Differential Equations, Paper No. 216, 17. [49] R. Wen, S. Chai and B.-Z. Guo, Well-posedness and exact controllability of fourth order Schrödinger equation with boundary control and collocated observation, SIAM J. Control Optim., 52 (2014), 365-396. doi: 10.1137/120902744. [50] R. Wen, S. Chai and B.-Z. Guo, Well-posedness and exact controllability of fourth-order Schrödinger equation with hinged boundary control and collocated observation, Math. Control Signals Systems, 28 (2016), Art. 22, 28. doi: 10.1007/s00498-016-0175-4. [51] J. Zhang and J. Zheng, Energy critical fourth-order Schrödinger equations with subcritical perturbations, Nonlinear Anal., 73 (2010), 1004-1014. doi: 10.1016/j.na.2010.04.027. [52] J. Zheng, Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity, Adv. Differential Equations, 16 (2011), 467-486. [53] S. Zhu, H. Yang and J. Zhang, Blow-up of rough solutions to the fourth-order nonlinear Schrödinger equation, Nonlinear Anal., 74 (2011), 6186-6201. doi: 10.1016/j.na.2011.05.096. [54] S. Zhu, J. Zhang and H. Yang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205. doi: 10.4310/DPDE.2010.v7.n2.a4. [55] S. Zhu, J. Zhang and H. Yang, Biharmonic nonlinear Schrödinger equation and the profile decomposition, Nonlinear Anal., 74 (2011), 6244-6255. doi: 10.1016/j.na.2011.06.004.
The region $D = D^+\cup D^-$
Partitioning the boundary
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