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

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu = < i partial derivative(x)>(-1-2)(2), (t, x) is an element of R x R, u(0, x) = u(0)(x), x is an element of R, where L = partial derivative(t) + i < i partial derivative(x)> and < i partial derivative(x)> = <(1 - partial derivative(2)(x))over bar>. 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.