Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

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December 2021, 14(12): 4631-4642. doi: 10.3934/dcdss.2021136

Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

College of Mathematical Sciences, Harbin Engineering University, No. 145 Nantong Street, Harbin 150001, China

^* Corresponding author: Chao Yang

Received September 2021 Revised October 2021 Published December 2021 Early access November 2021

This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term $ V(x)|\varphi|^{p-1}\varphi $ in $ \mathbb{R}^n $. For the case $ p>1+\frac{4(1+\varepsilon_0)}{n} (0<\varepsilon_0<\frac{2}{n-2}) $, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case $ p<1+\frac{4}{n} $, we obtain the global existence of solution for any initial data in $ H^1 (\mathbb{R}^n) $.

Keywords: Inhomogeneous nonlinear Schrödinger equation, potential well, global existence, blowup.

Mathematics Subject Classification: Primary: 35Q55, 35G25.

Citation: Chao Yang. Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4631-4642. doi: 10.3934/dcdss.2021136

References:

[1]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York University, New York, 2003. doi: 10.1090/cln/010.
[2]	A. de Bouard and R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré, 6 (2005), 1157-1177. doi: 10.1007/s00023-005-0236-6.
[3]	G. Fibich and X.-P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Physica. D., 175 (2003), 96-108. doi: 10.1016/S0167-2789(02)00626-7.
[4]	R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, J. Math. Kyoto Univ., 45 (2005), 145-158. doi: 10.1215/kjm/1250282971.
[5]	T. S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana, 55 (2000), 835-842. doi: 10.1007/s12043-000-0051-z.
[6]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[7]	H. Guo and T. Wang, A note on sign-changing solutions for the Schrödinger Poisson systerm, Electron. Res. Arch., 28 (2020), 195-203. doi: 10.3934/era.2020013.
[8]	L. Huang and J. Chen, Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system, Electron. Res. Arch., 28 (2020), 383-404. doi: 10.3934/era.2020022.
[9]	T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaé Phys. Théor., 46 (1987), 113-129.
[10]	T. Kato, Nonlinear Schrödinger equations, Schrödinger Operators (Sonderborg, 1988), 218–263. Lecture Notes in Physics, 345, Springer, Berlin, (1989). doi: 10.1007/3-540-51783-9_22.
[11]	Y. Liu, X.-P. Wang and K. Wang, Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 2105-2122. doi: 10.1090/S0002-9947-05-03763-3.
[12]	F. Merle, Nonexistence of minimal blow-up solutions of equations iu_t = ∆u − k(x)\|u\|^4/N u in R^N, Ann. Inst. H. Poincaré Phys. Théor., 64 (1996), 33-85.
[13]	P. Y. H. Pang, H. Tang and Y. Wang, Blow-up solutions of inhomogeneous nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 26 (2006), 137-169. doi: 10.1007/s00526-005-0362-5.
[14]	Y. Wang, Global existence and blow up of solutions for the inhomogenoeous nonlinear Schrödinger equation in $\mathbb{R}^2$, J. Math. Anal. Appl., 338 (2008), 1008-1019. doi: 10.1016/j.jmaa.2007.05.057.
[15]	M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894. doi: 10.1515/anona-2020-0031.
[16]	X. Zhao and W. Yan, Existence of standing waves for quasi-linear Schrödinger equations on $T^n$, Adv. Nonlinear Anal., 9 (2020), 978-993. doi: 10.1515/anona-2020-0038.

show all references

References:

[1]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York University, New York, 2003. doi: 10.1090/cln/010.
[2]	A. de Bouard and R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré, 6 (2005), 1157-1177. doi: 10.1007/s00023-005-0236-6.
[3]	G. Fibich and X.-P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Physica. D., 175 (2003), 96-108. doi: 10.1016/S0167-2789(02)00626-7.
[4]	R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, J. Math. Kyoto Univ., 45 (2005), 145-158. doi: 10.1215/kjm/1250282971.
[5]	T. S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana, 55 (2000), 835-842. doi: 10.1007/s12043-000-0051-z.
[6]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[7]	H. Guo and T. Wang, A note on sign-changing solutions for the Schrödinger Poisson systerm, Electron. Res. Arch., 28 (2020), 195-203. doi: 10.3934/era.2020013.
[8]	L. Huang and J. Chen, Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system, Electron. Res. Arch., 28 (2020), 383-404. doi: 10.3934/era.2020022.
[9]	T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaé Phys. Théor., 46 (1987), 113-129.
[10]	T. Kato, Nonlinear Schrödinger equations, Schrödinger Operators (Sonderborg, 1988), 218–263. Lecture Notes in Physics, 345, Springer, Berlin, (1989). doi: 10.1007/3-540-51783-9_22.
[11]	Y. Liu, X.-P. Wang and K. Wang, Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 2105-2122. doi: 10.1090/S0002-9947-05-03763-3.
[12]	F. Merle, Nonexistence of minimal blow-up solutions of equations iu_t = ∆u − k(x)\|u\|^4/N u in R^N, Ann. Inst. H. Poincaré Phys. Théor., 64 (1996), 33-85.
[13]	P. Y. H. Pang, H. Tang and Y. Wang, Blow-up solutions of inhomogeneous nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 26 (2006), 137-169. doi: 10.1007/s00526-005-0362-5.
[14]	Y. Wang, Global existence and blow up of solutions for the inhomogenoeous nonlinear Schrödinger equation in $\mathbb{R}^2$, J. Math. Anal. Appl., 338 (2008), 1008-1019. doi: 10.1016/j.jmaa.2007.05.057.
[15]	M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894. doi: 10.1515/anona-2020-0031.
[16]	X. Zhao and W. Yan, Existence of standing waves for quasi-linear Schrödinger equations on $T^n$, Adv. Nonlinear Anal., 9 (2020), 978-993. doi: 10.1515/anona-2020-0038.

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