| 概要 |
The scattering problem for the two-dimensional nonlinear Klein–Gordon equation with exponential nonlinearity 𝒩(u) is studied. It has been shown that the scattering operator 𝑺 ∶ (f_−, g_−) ↦ (f_+, g_+)... is well defined on a neighborhood in the energy space ℋ ∶ = H^1(ℝ^2) ⊕ L^2(ℝ^2) of 0. In this paper, we establish inequalities for the smoothness and decay of the output data (f_+, g_+) dependent on the smoothness and decay of the input data (f_−, g_−) and the smoothness of 𝒩. The inequalities imply that (f_+, g_+) belongs to the Schwartz space 𝒮(ℝ^2) ⊕ 𝒮(ℝ^2) if (f_−,g_−) is in 𝒮(ℝ^2) ⊕ 𝒮(ℝ^2) and small in the sense of ℋ, and that we have a simple sufficient condition for (f_−, g_−) and 𝒩such that all order partial derivatives of f_+ and g_+ decay more rapidly than a similar exponential function.続きを見る
|
Mendeley出力
