Final state problem for the cubic nonlinear Klein-Gordon equation

We study the final state problem for the nonlinear Klein-Gordon equation, u_tt +u- u_xx =μ u³, t ∈ R,x ∈ R, where μR. We prove the existence of solutions in the neighborhood of the approximate solutions 2 Re U (t) w+ (t), where U (t) is the free evolution group defined by U (t) = F^-1 e-^it 〈φ〉 F, 〈x〉 = 1+ x², F and F-1 are the direct and inverse Fourier transformations, respectively, and w+ (t,x) = F-1 (û₊ (φ) e^{(3/2) iμ 〈φ〉2} u₊ (φ) ² log t), with a given final data u+ is a real-valued function and ∥ 〈φ〉³ 〈 i∥_φ〉 u + (φ) ∥ L^∞ is small.