Wave operators to a quadratic nonlinear Klein–Gordon equation in two space dimensions Original Research Article

We study asymptotics around the final states of solutions to the nonlinear Klein–Gordon equations with quadratic nonlinearities in two space dimensions View the MathML source(∂t2−Δ+m2)u=λu2,(t,x)∈R×R2, where View the MathML sourceλ∈C. We prove that if the final states View the MathML sourceu1+∈Hqq−14−4q(R2)∩H52,1(R2)∩H12(R2), Turn MathJax on View the MathML sourceu2+∈Hqq−13−4q(R2)∩H32,1(R2)∩H11(R2), Turn MathJax on and View the MathML source‖u1+‖H12+‖u2+‖H11 Turn MathJax on is sufficiently small, where 4