Type: Article

Quadratic nonlinear Klein-Gordon equation in one dimension

Hayashi, Nakao; Naumkin, Pavel I.;

Abstract:

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation v(tt) + v - v(xx) = lambda v(2), t is an element of R, x is an element of R, with initial conditions v(0, x) = v(0)(x), v(t)(0, x) = v(1)(x), x is an element of R, where v(0) and v(0), are real-valued functions. lambda is an element of 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'equation de Klein-Gordon quasi-lineaire donnees petites en dimension I,"