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Decay rate of the Timoshenko system with one boundary damping
Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods
Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama city, Kanagawa, JAPAN |
We solve the scattering problems for nonlinear Schrödinger equations with an inverse-square potential by applying the energy methods. The methods are optimized to the abstract semilinear Schrödinger evolution equations with nonautonomous terms.
References:
[1] |
L. Baudouin, O. Kavian and J.-P. Puel,
Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations, 216 (2005), 188-222.
doi: 10.1016/j.jde.2005.04.006. |
[2] |
N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh,
Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.
doi: 10.1016/S0022-1236(03)00238-6. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, 2003.
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University
Press, New York, 1998. |
[5] |
T. Cazenave and F. B. Weissler,
Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
[6] |
J. Ginibre and T. Ozawa,
Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n \ge 2$, Comm. Math. Phys., 151 (1993), 619-645.
doi: 10.1007/BF02097031. |
[7] |
J. Ginibre and G. Velo,
Long range scattering and modified wave operators for some Hartree type equations. Ⅰ, Rev. Math. Phys., 12 (2000), 361-429.
doi: 10.1142/S0129055X00000137. |
[8] |
J. Ginibre and G. Velo,
Long range scattering and modified wave operators for some Hartree type equations. Ⅱ, Ann. Henri Poincaré, 1 (2000), 753-800.
doi: 10.1007/PL00001014. |
[9] |
N. Hayashi and T. Ozawa,
Scattering theory in the weighted $L^{2}(\mathbb{R}^{n})$ spaces for some Schrödinger equations, Ann. Inst. Henri Poincaré, 48 (1988), 17-37.
|
[10] |
N. Hayashi and Y. Tsutsumi,
Scattering theory for Hartree type equations, Ann. Inst. Henri Poincaré, 46 (1987), 187-213.
|
[11] |
J. Lu, C. Miao and J. Murphy,
Scattering in $H^{1}$ for the intercritical NLS with an inverse-square potential, J. Differ. Equ., 264 (2018), 3174-3211.
doi: 10.1016/j.jde.2017.11.015. |
[12] |
H. Mizutani,
Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, J. Differential Equations, 263 (2017), 3832-3853.
doi: 10.1016/j.jde.2017.05.006. |
[13] |
K. Nakanishi,
Modified wave operators for the Hartree equation with data, image and convergence in the same space. Ⅱ, Ann. Henri Poincaré, 3 (2002), 503-535.
doi: 10.1007/s00023-002-8626-5. |
[14] |
K. Nakanishi,
Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal., 1 (2002), 237-252.
doi: 10.3934/cpaa.2002.1.237. |
[15] |
N. Okazawa, T. Suzuki and T. Yokota,
Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629.
doi: 10.1080/00036811.2011.631914. |
[16] |
N. Okazawa, T. Suzuki and T. Yokota,
Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.
doi: 10.3934/eect.2012.1.337. |
[17] |
N. Okazawa, T. Yokota and K. Yoshii,
Remarks on linear Schrödinger evolution equations with Coulomb potential with moving center, SUT J. Math., 46 (2010), 155-176.
|
[18] |
N. Okazawa and K. Yoshii,
Linear Schrödinger evolution equations with moving Coulomb singularities, J. Differential Equations, 254 (2013), 2964-2999.
doi: 10.1016/j.jde.2013.01.017. |
[19] |
V. Pierfelice,
Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392.
|
[20] |
T. Suzuki,
Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542.
doi: 10.3934/eect.2013.2.531. |
[21] |
T. Suzuki,
Blowup of nonlinear Schrödinger equations with inverse-square potentials, Differ. Equ. Appl., 6 (2014), 309-333.
doi: 10.7153/dea-06-17. |
[22] |
T. Suzuki,
Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.
doi: 10.1619/fesi.59.1. |
[23] |
T. Suzuki,
Scattering theory for Hartree equations with inverse-square potentials, Appl. Anal., 96 (2017), 2032-2043.
doi: 10.1080/00036811.2016.1200720. |
[24] |
T. Suzuki,
Virial identities for nonlinear Schrödinger equations with an inverse-square potential of critical coefficient, Differ. Equ. Appl., 9 (2017), 327-352.
doi: 10.7153/dea-2017-09-24. |
[25] |
K. Yajima,
Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
doi: 10.1007/BF01212420. |
[26] |
J. Zhang and J. Zheng,
Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.
doi: 10.1016/j.jfa.2014.08.012. |
show all references
References:
[1] |
L. Baudouin, O. Kavian and J.-P. Puel,
Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations, 216 (2005), 188-222.
doi: 10.1016/j.jde.2005.04.006. |
[2] |
N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh,
Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549.
doi: 10.1016/S0022-1236(03)00238-6. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, 2003.
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University
Press, New York, 1998. |
[5] |
T. Cazenave and F. B. Weissler,
Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
[6] |
J. Ginibre and T. Ozawa,
Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n \ge 2$, Comm. Math. Phys., 151 (1993), 619-645.
doi: 10.1007/BF02097031. |
[7] |
J. Ginibre and G. Velo,
Long range scattering and modified wave operators for some Hartree type equations. Ⅰ, Rev. Math. Phys., 12 (2000), 361-429.
doi: 10.1142/S0129055X00000137. |
[8] |
J. Ginibre and G. Velo,
Long range scattering and modified wave operators for some Hartree type equations. Ⅱ, Ann. Henri Poincaré, 1 (2000), 753-800.
doi: 10.1007/PL00001014. |
[9] |
N. Hayashi and T. Ozawa,
Scattering theory in the weighted $L^{2}(\mathbb{R}^{n})$ spaces for some Schrödinger equations, Ann. Inst. Henri Poincaré, 48 (1988), 17-37.
|
[10] |
N. Hayashi and Y. Tsutsumi,
Scattering theory for Hartree type equations, Ann. Inst. Henri Poincaré, 46 (1987), 187-213.
|
[11] |
J. Lu, C. Miao and J. Murphy,
Scattering in $H^{1}$ for the intercritical NLS with an inverse-square potential, J. Differ. Equ., 264 (2018), 3174-3211.
doi: 10.1016/j.jde.2017.11.015. |
[12] |
H. Mizutani,
Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential, J. Differential Equations, 263 (2017), 3832-3853.
doi: 10.1016/j.jde.2017.05.006. |
[13] |
K. Nakanishi,
Modified wave operators for the Hartree equation with data, image and convergence in the same space. Ⅱ, Ann. Henri Poincaré, 3 (2002), 503-535.
doi: 10.1007/s00023-002-8626-5. |
[14] |
K. Nakanishi,
Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal., 1 (2002), 237-252.
doi: 10.3934/cpaa.2002.1.237. |
[15] |
N. Okazawa, T. Suzuki and T. Yokota,
Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629.
doi: 10.1080/00036811.2011.631914. |
[16] |
N. Okazawa, T. Suzuki and T. Yokota,
Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.
doi: 10.3934/eect.2012.1.337. |
[17] |
N. Okazawa, T. Yokota and K. Yoshii,
Remarks on linear Schrödinger evolution equations with Coulomb potential with moving center, SUT J. Math., 46 (2010), 155-176.
|
[18] |
N. Okazawa and K. Yoshii,
Linear Schrödinger evolution equations with moving Coulomb singularities, J. Differential Equations, 254 (2013), 2964-2999.
doi: 10.1016/j.jde.2013.01.017. |
[19] |
V. Pierfelice,
Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392.
|
[20] |
T. Suzuki,
Energy methods for Hartree type equation with inverse-square potentials, Evol. Equ. Control Theory, 2 (2013), 531-542.
doi: 10.3934/eect.2013.2.531. |
[21] |
T. Suzuki,
Blowup of nonlinear Schrödinger equations with inverse-square potentials, Differ. Equ. Appl., 6 (2014), 309-333.
doi: 10.7153/dea-06-17. |
[22] |
T. Suzuki,
Solvability of nonlinear Schrödinger equations with some critical singular potential via generalized Hardy-Rellich inequalities, Funkcial. Ekvac., 59 (2016), 1-34.
doi: 10.1619/fesi.59.1. |
[23] |
T. Suzuki,
Scattering theory for Hartree equations with inverse-square potentials, Appl. Anal., 96 (2017), 2032-2043.
doi: 10.1080/00036811.2016.1200720. |
[24] |
T. Suzuki,
Virial identities for nonlinear Schrödinger equations with an inverse-square potential of critical coefficient, Differ. Equ. Appl., 9 (2017), 327-352.
doi: 10.7153/dea-2017-09-24. |
[25] |
K. Yajima,
Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
doi: 10.1007/BF01212420. |
[26] |
J. Zhang and J. Zheng,
Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.
doi: 10.1016/j.jfa.2014.08.012. |
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