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Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity
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 |
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) $.
References:
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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 iut = ∆u − k(x)|u|4/N u in RN, 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 iut = ∆u − k(x)|u|4/N u in RN, 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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