Weighted Sobolev regularity and failure of profile convergence associated with the scattering operator for the three-dimensional cubic Klein-Gordon equation
Abstract
The scattering problem for the three-dimensional nonlinear Klein-Gordon equation utt−Δu+u=± u3 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-∈𝒮⊕𝒮 whose ℋ1/2-norm is sufficiently small, the corresponding solution u satisfies U(−t) t(u(t),∂t u(t)) not→ 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.
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BibTeX
@misc{sasaki2026weighted,
title = {{Weighted Sobolev regularity and failure of profile convergence associated with the scattering operator for the three-dimensional cubic Klein-Gordon equation}},
author = {Hironobu Sasaki},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02707},
eprint = {2611.02707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02707},
}
RIS
TY - UNPB
TI - Weighted Sobolev regularity and failure of profile convergence associated with the scattering operator for the three-dimensional cubic Klein-Gordon equation
AU - Hironobu Sasaki
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02707
UR - https://arxiv.org/abs/2611.02707
AB - The scattering problem for the three-dimensional nonlinear Klein-Gordon equation u_tt−Δu+u=± u^3 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 ℋ^1/2:=H^1/2(ℝ^3) ⊕ H^−1/2(ℝ^3), and that if Φ_- is sufficiently small in ℋ^1/2 and belongs to 𝒮 ⊕ 𝒮, then SΦ_- is also in 𝒮 ⊕ 𝒮, where 𝒮=𝒮(ℝ^3) is the Schwartz space. In the present paper, however, we prove that for any non-zero real-valued Φ_-∈𝒮⊕𝒮 whose ℋ^1/2-norm is sufficiently small, the corresponding solution u satisfies U(−t) ^t(u(t),∂_t u(t)) not→ SΦ_- (t→+∞) in the weighted Sobolev space H^1,3(ℝ^3)⊕ H^0,3(ℝ^3) although U(−t) ^t(u(t),∂_t u(t)) converges to SΦ_- in H^1,σ(ℝ^3)⊕ H^0,σ(ℝ^3) for any σ<3. Here, U(t) denotes the free Klein-Gordon evolution group.
ER -