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

Abstract EN

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.