Advances in Mathematical Physics
Volume 2010 (2010), Article ID 504324, 35 pages
Research Article

The Initial Value Problem for the Quadratic Nonlinear Klein-Gordon Equation

1Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan
2Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Mexico

Received 18 September 2009; Accepted 23 February 2010

Academic Editor: Dongho Chae

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.


We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu=ix-1u¯2, (t,x)R×R, u(0,x)=u0(x), xR, where L=t+iix and ix=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.