2009, 8(5): 1541-1554. doi: 10.3934/cpaa.2009.8.1541

Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential

1. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068

Received  February 2008 Revised  December 2008 Published  April 2009

In this paper, we apply a cross-constrained variational approach for the nonlinear Klein-Gordon equations with an inverse square potential in three space dimensions (which is a representative of the class of equations of interest) based on the relationship between a type of cross-constrained variational problem and energy. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we first derive a sharp threshold for global existence and blow-up of solutions to the Cauchy problem for the equations under study. On the other hand, we get an answer of the question: how small are the initial data, the global solutions exist?
Citation: Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[1]

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

[2]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071

[3]

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

[4]

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

[5]

Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711

[16]

Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753

[17]

Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525

[18]

Necdet Bildik, Sinan Deniz. New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 503-518. doi: 10.3934/dcdss.2020028

[19]

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

[20]

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

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]