We consider the initial value problem for systems of nonlinear Klein–Gordon equations with quadratic nonlinearities. We prove the existence of scattering states, namely, the asymptotic stability of small solutions in the neighborhood of the free solutions for small initial data in the weighted Sobolev space H4,3(R3)×H3,3(R3). If nonlinearities satisfy the strong null condition, then the same result is true in two space dimensions for small initial data in H5,4(R2)×H4,4(R2). A system of massive Dirac–massless Klein–Gordon equations in three space dimensions is also considered by our method.

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AbstractLet f be analytic in {\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}} with f(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n}, and for α ≥ 0 and 0 < λ ≤ 1, let { {\mathcal B} }_{1}(\alpha ,\lambda ) denote the subclass of Bazilevič functions satisfying \left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f\in { {\mathcal B} }_{1}(\alpha ,\lambda ), thus extending recent work in the case λ = 1.

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