January  2020, 40(1): 597-608. doi: 10.3934/dcds.2020024

Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

* Corresponding author: Ihyeok Seo

Received  April 2019 Published  October 2019

Fund Project: This research was supported by NRF-2019R1F1A1061316

We prove weighted $ L^2 $ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.

Citation: Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024
References:
[1]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math, 30 (1976), 1-38.  doi: 10.1007/BF02786703.  Google Scholar

[2]

J. A. BarcelòJ. M. BennettA. CarberyA. 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.  Google Scholar

[3]

P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z, 186 (1984), 383-391.  doi: 10.1007/BF01174891.  Google Scholar

[4]

S. Chanillo and E. Sawyer, Unique continuation for $\Delta + v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.  doi: 10.2307/2001239.  Google Scholar

[5]

F. Chiarenza and A. Ruiz, Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64.  doi: 10.2307/2048479.  Google Scholar

[6]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.  Google Scholar

[7]

P. D'Ancona, Kato smoothing and Strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys., 335 (2015), 1-16.  doi: 10.1007/s00220-014-2169-8.  Google Scholar

[8]

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.  Google Scholar

[9]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.  doi: 10.1080/03605300701743749.  Google Scholar

[10]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.  doi: 10.1007/BF01360915.  Google Scholar

[11]

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.  Google Scholar

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.   Google Scholar
[13]

A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J., 76 (1994), 913-940.  doi: 10.1215/S0012-7094-94-07636-9.  Google Scholar

[14]

I. Seo, From resolvent estimates to unique continuation for the Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 8755-8784.  doi: 10.1090/tran/6635.  Google Scholar

[15] C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511530029.  Google Scholar
[16]

R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[17]

J. Zhang and J. Zheng, Strichartz estimate and nonlinear Klein-Gordon equation on nontrapping scattering space, J. Geom. Anal., 29 (2019), 2957-2984.  doi: 10.1007/s12220-018-00100-3.  Google Scholar

show all references

References:
[1]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math, 30 (1976), 1-38.  doi: 10.1007/BF02786703.  Google Scholar

[2]

J. A. BarcelòJ. M. BennettA. CarberyA. 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.  Google Scholar

[3]

P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z, 186 (1984), 383-391.  doi: 10.1007/BF01174891.  Google Scholar

[4]

S. Chanillo and E. Sawyer, Unique continuation for $\Delta + v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.  doi: 10.2307/2001239.  Google Scholar

[5]

F. Chiarenza and A. Ruiz, Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64.  doi: 10.2307/2048479.  Google Scholar

[6]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.  doi: 10.1006/jfan.2000.3687.  Google Scholar

[7]

P. D'Ancona, Kato smoothing and Strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys., 335 (2015), 1-16.  doi: 10.1007/s00220-014-2169-8.  Google Scholar

[8]

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.  Google Scholar

[9]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.  doi: 10.1080/03605300701743749.  Google Scholar

[10]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279.  doi: 10.1007/BF01360915.  Google Scholar

[11]

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.  Google Scholar

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.   Google Scholar
[13]

A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J., 76 (1994), 913-940.  doi: 10.1215/S0012-7094-94-07636-9.  Google Scholar

[14]

I. Seo, From resolvent estimates to unique continuation for the Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 8755-8784.  doi: 10.1090/tran/6635.  Google Scholar

[15] C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511530029.  Google Scholar
[16]

R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[17]

J. Zhang and J. Zheng, Strichartz estimate and nonlinear Klein-Gordon equation on nontrapping scattering space, J. Geom. Anal., 29 (2019), 2957-2984.  doi: 10.1007/s12220-018-00100-3.  Google Scholar

[1]

Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771

[2]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903

[3]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[4]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[5]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

[6]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233

[7]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973

[8]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[9]

Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085

[10]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251

[11]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215

[12]

Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723

[13]

Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143

[14]

Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134

[15]

Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050

[16]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100

[17]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235

[18]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

[19]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

[20]

Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (33)
  • HTML views (52)
  • Cited by (0)

Other articles
by authors

[Back to Top]