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The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition
On Morawetz estimates with time-dependent weights for the Klein-Gordon equation
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea |
$ \begin{equation*} \big\| |\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^{2}_{x, t}(|(x, t)|^{-\alpha})} \lesssim \| f \|_{H^s} \end{equation*} $ |
$ \sigma, s\geq0 $ |
$ \alpha>0 $ |
$ |x|^{-\alpha} $ |
$ |(x, t)|^{-\alpha} $ |
$ A_2 $ |
$ \sqrt{1-\Delta} $ |
References:
[1] |
J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela,
A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut, 21 (2008), 481-488.
doi: 10.5209/rev_REMA.2008.v21.n2.16405. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-66451-9. |
[3] |
P. D'Ancona,
On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations, Commun. Pure Appl. Anal., 19 (2020), 609-640.
doi: 10.3934/cpaa.2020029. |
[4] |
L. Grafakos, Classical Fourier Analysis, 3$^{rd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2014.
doi: 10.1007/978-1-4939-1194-3. |
[5] |
T. Kato,
Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279.
doi: 10.1007/BF01360915. |
[6] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in applied mathematics, Adv. Math. Suppl. Stud., Academic Press, New York, 8 (1983), 93–128. |
[7] |
T. Kato and K. Yajima,
Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.
doi: 10.1142/S0129055X89000171. |
[8] |
Y. Koh and I. Seo,
Global well-posedness for higher-order Schrödinger equations in weighted $L^2$ spaces, Comm. Partial Differential Equations, 40 (2015), 1815-1830.
doi: 10.1080/03605302.2015.1048551. |
[9] |
Y. Koh and I. Seo,
On weighted $L^2$ estimates for solutions of the wave equation, Proc. Amer. Math. Soc., 144 (2016), 3047-3061.
doi: 10.1090/proc/12951. |
[10] |
Y. Koh and I. Seo,
Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications, Discrete Contin. Dyn. Syst., 37 (2017), 4877-4906.
doi: 10.3934/dcds.2017210. |
[11] |
Y. Koh and I. Seo, Strichartz and smoothing estimates in weighted $L^2$ spaces and their applications, to appear in Indiana Univ. Math. J., arXiv: 1803.10430. |
[12] |
D. S. Kurtz,
Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254.
doi: 10.2307/1998156. |
[13] |
H. Lee, I. Seo and J. Seok,
Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential, Discrete Contin. Dyn. Syst., 40 (2020), 597-608.
doi: 10.3934/dcds.2020024. |
[14] |
W. Littman,
Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770.
doi: 10.1090/S0002-9904-1963-11025-3. |
[15] |
C. S. Morawetz,
Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. London Ser. A, 306 (1968), 291-296.
doi: 10.1098/rspa.1968.0151. |
[16] |
T. Ozawa and K. M. Rogers,
Sharp Morawetz estimates, J. Anal. Math., 121 (2013), 163-175.
doi: 10.1007/s11854-013-0031-0. |
[17] |
M. Ruzhansky and M. Sugimoto,
Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. Lond. Math. Soc., 105 (2012), 393-423.
doi: 10.1112/plms/pds006. |
[18] |
I. Seo,
A note on the Schrödinger smoothing effect, Math. Nachr., 292 (2019), 2481-2487.
doi: 10.1002/mana.201800502. |
[19] |
E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
doi: 10.1515/9781400883929. |
show all references
References:
[1] |
J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela,
A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut, 21 (2008), 481-488.
doi: 10.5209/rev_REMA.2008.v21.n2.16405. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-66451-9. |
[3] |
P. D'Ancona,
On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations, Commun. Pure Appl. Anal., 19 (2020), 609-640.
doi: 10.3934/cpaa.2020029. |
[4] |
L. Grafakos, Classical Fourier Analysis, 3$^{rd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2014.
doi: 10.1007/978-1-4939-1194-3. |
[5] |
T. Kato,
Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279.
doi: 10.1007/BF01360915. |
[6] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in applied mathematics, Adv. Math. Suppl. Stud., Academic Press, New York, 8 (1983), 93–128. |
[7] |
T. Kato and K. Yajima,
Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.
doi: 10.1142/S0129055X89000171. |
[8] |
Y. Koh and I. Seo,
Global well-posedness for higher-order Schrödinger equations in weighted $L^2$ spaces, Comm. Partial Differential Equations, 40 (2015), 1815-1830.
doi: 10.1080/03605302.2015.1048551. |
[9] |
Y. Koh and I. Seo,
On weighted $L^2$ estimates for solutions of the wave equation, Proc. Amer. Math. Soc., 144 (2016), 3047-3061.
doi: 10.1090/proc/12951. |
[10] |
Y. Koh and I. Seo,
Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications, Discrete Contin. Dyn. Syst., 37 (2017), 4877-4906.
doi: 10.3934/dcds.2017210. |
[11] |
Y. Koh and I. Seo, Strichartz and smoothing estimates in weighted $L^2$ spaces and their applications, to appear in Indiana Univ. Math. J., arXiv: 1803.10430. |
[12] |
D. S. Kurtz,
Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254.
doi: 10.2307/1998156. |
[13] |
H. Lee, I. Seo and J. Seok,
Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential, Discrete Contin. Dyn. Syst., 40 (2020), 597-608.
doi: 10.3934/dcds.2020024. |
[14] |
W. Littman,
Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770.
doi: 10.1090/S0002-9904-1963-11025-3. |
[15] |
C. S. Morawetz,
Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. London Ser. A, 306 (1968), 291-296.
doi: 10.1098/rspa.1968.0151. |
[16] |
T. Ozawa and K. M. Rogers,
Sharp Morawetz estimates, J. Anal. Math., 121 (2013), 163-175.
doi: 10.1007/s11854-013-0031-0. |
[17] |
M. Ruzhansky and M. Sugimoto,
Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. Lond. Math. Soc., 105 (2012), 393-423.
doi: 10.1112/plms/pds006. |
[18] |
I. Seo,
A note on the Schrödinger smoothing effect, Math. Nachr., 292 (2019), 2481-2487.
doi: 10.1002/mana.201800502. |
[19] |
E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
doi: 10.1515/9781400883929. |
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