Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-19T10:31:07.140Z Has data issue: false hasContentIssue false
  • English
  • Français

CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS*

Published online by Cambridge University Press:  01 September 2008

ZAIHUI GAN  and
JIAN ZHANG
ZAIHUI GAN
Affiliation:
College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, P.R. China e-mail: ganzaihui2008cn@yahoo.com.cn
JIAN ZHANG
Affiliation:
College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, P.R. China e-mail: ganzaihui2008cn@yahoo.com.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we put forward a cross-constrained variational method to study the non-linear Klein–Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

Keywords

35A1535L15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008 , pp. 467 - 481
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Ball, J. M., Finite time blow-up in nonlinear problems, in Nonlinear Evolution Equations (Grandall, M. G., Editor) (Academic press, New York, 1978), 189–205.Google Scholar
2.Berestycki, H. and Cazenave, T., Instabilité des états stationnaires dans les équations de Schrödinger et de KG nonlinéaires, C.K. Acad. Sci. Paris 293 (1981), 489–492.Google Scholar
3.Cazenave, T., An introduction to nonlinear Schrödinger equations, Textos de Metodos Mathematicos, vol. 22, Rio dé Janeiro: Instituto de Mathemática-UFRJ, 1989.Google Scholar
4.Ding, W. Y. and Ni, W. M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986), 283–308.CrossRefGoogle Scholar
5.Gan, Z. and Zhang, J., Standing waves of nonlinear Klein–Gordon equations with nonnegative potentials, Appl. Math. Comput. 166 (2005), 551–570.Google Scholar
6.Glassey, R. T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794–1797.CrossRefGoogle Scholar
7.Kavian, O., A remark on the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 193 (1) (1987), 193–203.Google Scholar
8.Kichenassamy, S., Breather solutions of nonlinear wave equations, Commun. Pure Appl. Math. 44 (1991), 789–818.CrossRefGoogle Scholar
9.Levine, H. A., Instability and non-existence of global solutions to nonlinear wave equations of the form Pu tt=−Au+F(u), Trans. Amer. Math. Soc. 192 (1974), 1–21.Google Scholar
10.Lindblad, H. and Sogge, C. D., On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357–426.CrossRefGoogle Scholar
11.Liu, Y., Existence and blow-up of solutions of a nonlinear Pochhammmer chree equation, Indian Univ. Math. J. 45 (3) (1996), 797–816.Google Scholar
12.Ni, W. M. and Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. XLIV (1991), 819–851.CrossRefGoogle Scholar
13.Payne, L. E. and Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (3–4) (1975), 273–303.CrossRefGoogle Scholar
14.Pecher, H., Low energy scattering for nonlinear Klein–Gordon equations, J. Funct. Anal. 63 (1985), 101–122.CrossRefGoogle Scholar
15.Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.CrossRefGoogle Scholar
16.Simon Jacques, C. H. and Taflin, E., The Cauchy problem for nonlinear Klein–Gordon equations, Comm. Math. Phys. 71 (1993), 433–445.CrossRefGoogle Scholar
17.Soffer, A. and Weinstein, M. I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9–74.CrossRefGoogle Scholar
18.Strauss, W., Nonlinear scattering theory at low energy, J. Funct. Anal. 42 (1981), 110–133 and 43 (1981), 281–293.CrossRefGoogle Scholar
19.Strauss, W., Nonlinear wave equations, C.B.M.S., Regional Conference Series in Mathematics, No. 73, 1989.CrossRefGoogle Scholar
20.Wang, X. F., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), 229–244.CrossRefGoogle Scholar
21.Wang, B. X., On existence and scattering for critical and subcritical nonlinear Klein–Gordon equations in H s, Nonlinear Anal. TMA. 31 (1998), 573–587.CrossRefGoogle Scholar
22.Wang, B. X. and Hudzik, H., The global Cauchy problem for the NLS and NLKG with small rough data, J. Diff. Eq. 231 (2007), 36–73.CrossRefGoogle Scholar
23.Zhang, J., Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys. 51 (2000), 498–503.CrossRefGoogle Scholar
24.Zhang, J., Sharp conditions of global existence for nonlinear Schrödinger and Klein–Gordon equations, Nonlinear Anal. TMA. 48 (2002), 191–267.CrossRefGoogle Scholar
25.Zhang, J., Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Commun. PDE 30 (2005), 1429–1443.CrossRefGoogle Scholar