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

Hyeongjin Lee; Ihyeok Seo; Jihyeon Seok

doi:10.3934/dcds.2020024

Article Contents

2020, Volume 40, Issue 1: 597-608. Doi: 10.3934/dcds.2020024

This issue Previous Article Existence and nonexistence of subsolutions for augmented Hessian equations Next Article Global stability of Keller–Segel systems in critical Lebesgue spaces

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
^* Corresponding author: Ihyeok Seo

Received: March 31, 2019

Published: October 2019

This research was supported by NRF-2019R1F1A1061316

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B45; Secondary: 35Q40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[2]	J. A. Barcelò, 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.
[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.
[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.
[5]	F. Chiarenza and A. Ruiz, Uniform $L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc., 112 (1991), 53-64. doi: 10.2307/2048479.
[6]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.
[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.
[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.
[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.
[10]	T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1966), 258-279. doi: 10.1007/BF01360915.
[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.
[12]	M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.
[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.
[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.
[15]	C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511530029.
[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.
[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.