Quadratic nonlinear Klein-Gordon equation in one dimension

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation v tt + v − v xx = λv 2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v 0(x), v t (0, x) = v 1(x), x ∈ R, where v 0 and v 1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah [“Normal for...

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Bibliographic Details
Published in:Journal of mathematical physics 2012-10, Vol.53 (10)
Main Authors: Hayashi, Nakao, Naumkin, Pavel I.
Format: Article
Language:English
Subjects:
Asymptotic properties
Exact sciences and technology
Initial conditions
Initial value problems
Klein-Gordon equation
Mathematical analysis
Mathematical methods in physics
Mathematics
Nonlinearity
Physics
Sciences and techniques of general use
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Summary:We study the initial value problem for the quadratic nonlinear Klein-Gordon equation v tt + v − v xx = λv 2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v 0(x), v t (0, x) = v 1(x), x ∈ R, where v 0 and v 1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah [“Normal forms and quadratic nonlinear Klein-Gordon equations,” Commun. Pure Appl. Math. 38, 685–696 (1985)], we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data, which was assumed in the previous work of J.-M. Delort [“Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi-linéaire á données petites en dimension 1,” Ann. Sci. Ec. Normale Super. 34(4), 1–61 (2001)].
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4759156