Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations

GAN Zai-Hui; ZHANG Jian

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GAN Zai-Hui, ZHANG Jian. Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations[J]. Applied Mathematics and Mechanics, 2005, 26(7): 833-839.

Citation:

GAN Zai-Hui, ZHANG Jian. Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations[J]. Applied Mathematics and Mechanics, 2005, 26(7): 833-839.

Citation:

GAN Zai-Hui, ZHANG Jian. Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations[J]. Applied Mathematics and Mechanics, 2005, 26(7): 833-839.

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Asymptotic Theory of Initial Value Problems for Nonlinear Perturbed Klein-Gordon Equations

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, P. R. China

Received Date: 2003-03-07
Rev Recd Date: 2005-04-26
Publish Date: 2005-07-15

Abstract

Abstract

The asymptotic theory of initial value problems for a class of nonlinear perturbed Klein- Gordon equations in two space dimensions is considered. Firstly, using the contraction mapping principle, combining some priori estimates and the convergence of Bessel function, the well-posedness of solutions of the initial value problem in twice continuous differentiable space was obtained according to the equivalent integral equation of initial value problem for the Klein-Gordon equations. Next, formal approximations of initial value problem was constructed by perturbation method and the asymptotic validity of the formal approximation is got. Finally, an application of the asymptotic theory was given, the asymptotic approximation degree of solutions for the initial value problem of a specific nonlinear Klein-Gordon equation was analyzed by using the asymptotic approximation theorem.
- Klein-Gordon equations,
- well-posedness,
- asymptotic theory,
- formal approximations,
- application

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References(9)

References

[1]	Van Horssen W T.Asymptotics for a class of semilinear hyperbolic equations with an application to a problem with a quadratic nonlinearity[J].Non Anal TMA,1992,19(6):501—530. doi: 10.1016/0362-546X(92)90018-A
[2]	Van Horssen W T,Van Der Burgh A H.On initial boundary value problems for weakly semilinear telegraph equations. asymptotic theory and application[J].SIAM J Appl Math,1988,48(4):719—736. doi: 10.1137/0148041
[3]	WANG Bao-xiang. On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in Hs[J].Nonlinear Anal TMA,1998,31(5/6):573—587. doi: 10.1016/S0362-546X(97)00424-0
[4]	Pecher H.Lp-Abschtzungen und klassiche Lsungen für nichtlineare Wellengeichungen[J].I Math Z,1976,150(2):159—183. doi: 10.1007/BF01215233
[5]	Kapitanskii L V.Weak and yet weak solutions of semilinear wave equations[J].Comm Partial Diff Equations,1994,19(7):1629—1676. doi: 10.1080/03605309408821067
[6]	Pecher H.Nonlinear small data scattering for the wave and Klein-Gordon equations[J].Math Z,1984,185(3):261—270. doi: 10.1007/BF01181697
[7]	Pecher H.Low energy scattering for nonlineaar Klein-Gordon equations[J].J Functional Anal,1985,63(1):101—122. doi: 10.1016/0022-1236(85)90100-4
[8]	Guenther Ronald B,Lee John W.Partial Differential Equations of Mathematical Physics and Integral Equations[M].New Jersey:Prentice Hall,1988.
[9]	Kji Kubota.Existence of a global solution to semilinear wave equations with initial data of noncompact support in low space dimensions[J].Hokkaido Math,1993,22(1):123—180.