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A remark on norm inflation for nonlinear Schrödinger equations
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Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up
A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach
Department of Mathematics, Shimane University, Matsue 690-8504, Japan |
$\partial_t u +i \Delta u = i\lambda |u|^{p-1} u$ |
$\mathit{\boldsymbol{R}}^{1+n}$ |
$n\ge 3$ |
$p>1$ |
$\lambda \in \mathit{\boldsymbol{C}}$ |
$H^s$ |
$1<s<\min\{4;n/2\}$ |
$\max\{1;s/2\}< p< 1+4/(n-2s)$ |
$p$ |
References:
[1] |
H. Amann,
Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.
doi: 10.1002/mana.3211860102. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. |
[3] |
T. Cazenave, D. Fang and Z. Han,
Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 7911-7934.
doi: 10.1090/tran6683. |
[4] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[5] |
D. Fang and Z. Han,
On the well-posedness for NLS in $H^s$, J. Funct. Anal., 264 (2013), 1438-1455.
doi: 10.1016/j.jfa.2013.01.005. |
[6] |
J. Ginibre, T. Ozawa and G. Velo,
On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239.
|
[7] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[8] |
J. Ginibre and G. Velo,
The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
|
[9] |
J. Ginibre and G. Velo,
Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573.
|
[10] |
J. Ginibre and G. Velo,
Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.
doi: 10.1006/jfan.1995.1119. |
[11] |
T. Kato,
On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
|
[12] |
T. Kato, Nonlinear Schrödinger equations, in Schrödinger Operators, Lecture Notes in Phys., 345, Springer, Berlin (1989), 218–263.
doi: 10.1007/3-540-51783-9_22. |
[13] |
T. Kato,
On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.
doi: 10.1007/BF02787794. |
[14] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
|
[15] |
M. Nakamura and T. Ozawa,
Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys., 9 (1997), 397-410.
doi: 10.1142/S0129055X97000154. |
[16] |
M. Nakamura and T. Wada,
Modified Strichartz estimates with an application to the critical nonlinear Schrödinger equation, Nonlinear Anal., 130 (2016), 138-156.
doi: 10.1016/j.na.2015.09.023. |
[17] |
H. Pecher,
Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincaré Phys. Théor., 67 (1997), 259-296.
|
[18] |
H. Y. Schmeisser, Vector-valued Sobolev and Besov spaces, in Seminar Analysis of the KarlWeierstraß-Institute of Mathematics 1985/86 (Berlin, 1985/86), Teubner-Texte Math. 96, Teubner, Leipzig (1987), 4–44. |
[19] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam-New York-Oxford, 1978. |
[20] |
Y. Tsutsumi,
Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal., 11 (1987), 1143-1154.
doi: 10.1016/0362-546X(87)90003-4. |
[21] |
Y. Tsutsumi,
$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.
|
[22] |
H. Uchizono and T. Wada,
Continuous dependence for nonlinear Schrödinger equation in $H^s$, J. Math. Sci. Univ. Tokyo, 19 (2012), 57-68.
|
[23] |
H. Uchizono and T. Wada,
On well-posedness for nonlinear Schrödinger equations with power nonlinearity in fractional order Sobolev spaces, J. Math. Anal. Appl., 395 (2012), 56-62.
doi: 10.1016/j.jmaa.2012.04.079. |
show all references
References:
[1] |
H. Amann,
Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.
doi: 10.1002/mana.3211860102. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. |
[3] |
T. Cazenave, D. Fang and Z. Han,
Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 7911-7934.
doi: 10.1090/tran6683. |
[4] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[5] |
D. Fang and Z. Han,
On the well-posedness for NLS in $H^s$, J. Funct. Anal., 264 (2013), 1438-1455.
doi: 10.1016/j.jfa.2013.01.005. |
[6] |
J. Ginibre, T. Ozawa and G. Velo,
On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239.
|
[7] |
J. Ginibre and G. Velo,
On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[8] |
J. Ginibre and G. Velo,
The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
|
[9] |
J. Ginibre and G. Velo,
Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573.
|
[10] |
J. Ginibre and G. Velo,
Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.
doi: 10.1006/jfan.1995.1119. |
[11] |
T. Kato,
On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
|
[12] |
T. Kato, Nonlinear Schrödinger equations, in Schrödinger Operators, Lecture Notes in Phys., 345, Springer, Berlin (1989), 218–263.
doi: 10.1007/3-540-51783-9_22. |
[13] |
T. Kato,
On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.
doi: 10.1007/BF02787794. |
[14] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
|
[15] |
M. Nakamura and T. Ozawa,
Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys., 9 (1997), 397-410.
doi: 10.1142/S0129055X97000154. |
[16] |
M. Nakamura and T. Wada,
Modified Strichartz estimates with an application to the critical nonlinear Schrödinger equation, Nonlinear Anal., 130 (2016), 138-156.
doi: 10.1016/j.na.2015.09.023. |
[17] |
H. Pecher,
Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincaré Phys. Théor., 67 (1997), 259-296.
|
[18] |
H. Y. Schmeisser, Vector-valued Sobolev and Besov spaces, in Seminar Analysis of the KarlWeierstraß-Institute of Mathematics 1985/86 (Berlin, 1985/86), Teubner-Texte Math. 96, Teubner, Leipzig (1987), 4–44. |
[19] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam-New York-Oxford, 1978. |
[20] |
Y. Tsutsumi,
Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal., 11 (1987), 1143-1154.
doi: 10.1016/0362-546X(87)90003-4. |
[21] |
Y. Tsutsumi,
$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.
|
[22] |
H. Uchizono and T. Wada,
Continuous dependence for nonlinear Schrödinger equation in $H^s$, J. Math. Sci. Univ. Tokyo, 19 (2012), 57-68.
|
[23] |
H. Uchizono and T. Wada,
On well-posedness for nonlinear Schrödinger equations with power nonlinearity in fractional order Sobolev spaces, J. Math. Anal. Appl., 395 (2012), 56-62.
doi: 10.1016/j.jmaa.2012.04.079. |
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