American Institute of Mathematical Sciences

Advanced
November  2003, 9(6): 1387-1400. doi: 10.3934/dcds.2003.9.1387

Dispersive estimate for the wave equation with the inverse-square potential

Fabrice Planchon 1, , John G. Stalker 2, and  A. Shadi Tahvildar-Zadeh 3,

1. 

Laboratoire Analyse, Géométrie & Applications, UMR 7539, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France
2. 

Department of Mathematics, Princeton University, Princeton N.J. 08544, United States
3. 

Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway NJ 08854, United States

Received  January 2002 Revised  March 2003 Published  September 2003

We prove that spherically symmetric solutions of the Cauchy problem for the linear wave equation with the inverse-square potential satisfy a modified dispersive inequality that bounds the $L^\infty$ norm of the solution in terms of certain Besov norms of the data, with a factor that decays in $t$ for positive potentials. When the potential is negative we show that the decay is split between $t$ and $r$, and the estimate blows up at $r=0$. We also provide a counterexample showing that the use of Besov norms in dispersive inequalities for the wave equation are in general unavoidable.
Keywords: inverse-square potential, dispersive estimates, Wave equation, time decay..
Mathematics Subject Classification: 35L05, 35L1.
Citation: Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[1]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[2]

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
[3]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623
[4]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011
[5]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073
[6]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162
[7]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
[8]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022
[9]

Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[10]

Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65
[11]

Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531
[12]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074
[13]

Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019
[14]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649
[15]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[16]

Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195
[17]

Min Chen, S. Dumont, Louis Dupaigne, Olivier Goubet. Decay of solutions to a water wave model with a nonlocal viscous dispersive term. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1473-1492. doi: 10.3934/dcds.2010.27.1473
[18]

Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151
[19]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019
[20]

Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences