Weighted Sobolev regularity and failure of profile convergence associated with the scattering operator for the three-dimensional cubic Klein-Gordon equation

The scattering problem for the three-dimensional nonlinear Klein-Gordon equation {align*} uₜₜ-Δu+u=± u^3 {align*} is studied. It was shown that the scattering operator S for the equation is well-defined on a neighborhood of 0 in the Sobolev space H1/2:=H1/2(R³) ⊕ H-1/2(R³), and that if Φ₋ is sufficiently small in H1/2 and belongs to S ⊕ S, then SΦ₋ is also in S ⊕ S, where S=S(R³) is the Schwartz space. In the present paper, however, we prove that for any non-zero real-valued Φ₋ whose H1/2-norm is sufficiently small, the corresponding solution u satisfies \[ U(-t) \,{}^t(u(t),∂_t u(t)) → SΦ_- (t→+∞) \] in the weighted Sobolev space H1,3(R³)⊕ H0,3(R³) although U(-t)\,ᵗ(u(t),∂ₜ u(t)) converges to SΦ₋ in H1,σ(R³)⊕ H0,σ(R³) for any $σ<3$. Here, $U(t)$ denotes the free Klein-Gordon evolution group.