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

Abstract

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation ℒ𝑢=⟨𝑖𝜕𝑥⟩−1𝑢2, (𝑡,𝑥)∈𝐑×𝐑, 𝑢(0,𝑥)=𝑢0(𝑥), 𝑥∈𝐑, where ℒ=𝜕𝑡+𝑖⟨𝑖𝜕𝑥⟩ and ⟨𝑖𝜕𝑥⟩=1−𝜕2𝑥. 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.