Pecher, Hartmut Low energy scttering for nonlinear Klein-Gordon equations. (English) Zbl 0588.35061 J. Funct. Anal. 63, 101-122 (1985). The semilinear Klein-Gordon equation \[ u_{tt}-\Delta u+m^ 2u+f(u)=0\quad (m\neq 0) \] in arbitrary space dimension \(n\geq 2\) is considered as a perturbation of the linear Klein-Gordon equation. It is shown that in the case of the Cauchy problem the scattering operator exists in a whole neighbourhood of the origin in energy space, provided f behaves like a power \(| u|^{\rho}\) with \(1+4/(n-1)<\rho <1+4/(n-2)\) \((\rho <\infty\) for \(n=2)\). This extends the range \(1+4/n\leq \rho \leq 1+4/n-1\) which was obtained by W. A. Strauss before [J. Funct. Anal. 41, 110-133 (1981; Zbl 0466.47006)]. The proof consists of using weak decay estimates for the linear problem in the case of finite energy data combined with Schauder’s fixed point theorem. Cited in 38 Documents MSC: 35L70 Second-order nonlinear hyperbolic equations 47H10 Fixed-point theorems 35L05 Wave equation 35B40 Asymptotic behavior of solutions to PDEs Keywords:Klein-Gordon equation; perturbation; Cauchy problem; scattering operator; weak decay estimates; Schauder’s fixed point theorem Citations:Zbl 0466.47006 PDFBibTeX XMLCite \textit{H. Pecher}, J. Funct. Anal. 63, 101--122 (1985; Zbl 0588.35061) Full Text: DOI References: [1] Bergh, J.; Löfström, J., Interpolation Spaces (1976), Springer: Springer Berlin/Heidelberg/New York · Zbl 0344.46071 [3] Brenner, P., On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186, 383-391 (1984) · Zbl 0524.35084 [4] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin/Heidelberg/New York · Zbl 0691.35001 [5] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod, Gauthier-Villars: Dunod, Gauthier-Villars Paris · Zbl 0189.40603 [6] Marshall, B., Mixed norm estimates for the Klein-Gordon equation, (“Proceedings of a Conference on Harmonic Analysis in Honor A. Zygmund,” Chicago, 1981, Vol. 2 (1983), Springer: Springer Berlin), 638-645 [7] Pecher, H., Nonlinear small data scattering for the wave and Klein — Gordon equation, Math. Z., 185, 261-270 (1984) · Zbl 0538.35063 [8] Strauss, W., Nonlinear scattering theory at low energy, J. Funct. Anal., 41, 110-133 (1981) · Zbl 0466.47006 [9] Strauss, W., Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43, 281-293 (1981) · Zbl 0494.35068 [10] Strichartz, R. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44, 705-714 (1977) · Zbl 0372.35001 [11] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), North-Holland: North-Holland Amsterdam/New York/Oxford · Zbl 0387.46032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.