Quadratic nonlinear Klein-Gordon equation in one dimension

2012/10/01 by Nakao Hayashi, Pavel I. Naumkin · 1 citation
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations #Nonlinear Partial Differential Equations

paper · doi:10.1063/1.4759156

Abstract

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation vtt + v − vxx = λv2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v0(x), vt(0, x) = v1(x), x ∈ R, where v0 and v1 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)].

Citations

• Normal forms and quadratic nonlinear Klein‐Gordon equations

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