Abstract
We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We give some estimates for the nonlinearity, and prove the existence of the scattering operator, which improves the known results in some sense. Our proof is based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation.
Highlights
This paper is concerned with the scattering problem for the nonlinear
(f−, g− ) → (f+, g+ ) ∈ X s,0 if the following condition holds for some δ > 0: For any (f−, g− ) ∈ B(δ; X s,σ ), there uniquely exist a time-global solution u ∈ C(R; H s ) of (1.1), and data (f+, g+ ) ∈ X s,0 such that u(t) approaches u± (t) in H s as t tends to ±∞, where u± (t) are solutions of linear Klein-Gordon equations whose initial data are (f±, g± ), respectively
By using the Strichartz estimate for pre-admissible pair and the complex interpolation method for the weighted Sobolev space, we show that (S, X s,σ ) is well-defined if 4/3 < γ < 2 and σ > (2 − γ)/2, which improves the condition above
Summary
Asymptotic behavior, scattering, Klein-Gordon equation, cubic convolution. (f− , g− ) → (f+ , g+ ) ∈ X s,0 if the following condition holds for some δ > 0: For any (f− , g− ) ∈ B(δ; X s,σ ), there uniquely exist a time-global solution u ∈ C(R; H s ) of (1.1), and data (f+ , g+ ) ∈ X s,0 such that u(t) approaches u± (t) in H s as t tends to ±∞, where u± (t) are solutions of linear Klein-Gordon equations whose initial data are (f± , g± ), respectively. By using the Strichartz estimate for pre-admissible pair and the complex interpolation method for the weighted Sobolev space, we show that (S, X s,σ ) is well-defined if 4/3 < γ < 2 and σ > (2 − γ)/2, which improves the condition above. For recent results on the wave equation with a cubic convolution, see, e.g., Hidano [3] and Tsutaya [13]
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