Copyright © 2010 Nakao Hayashi and Pavel I. Naumkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu=〈i∂x〉-1u¯2, (t,x)∈R×R, u(0,x)=u0(x), x∈R, where L=∂t+i〈i∂x〉 and 〈i∂x〉=1-∂x2̅. Using the Shatah normal forms method, 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 works.