Scattering of solutions of nonlinear Klein-Gordon equations in three space dimensions
This paper is devoted to the theory of scattering for the non-linear Klein-Gordon equation \[ (*)\quad u_{tt}-\Delta u+m^ 2u+h(u)=0 \] in space dimension \(n=3\), with \(h\in {\mathcal C}^ 1\) and h satisfying a power estimate \(| h'(u)| \leq C| u|^{p-1}\) with \(3<p<5\). The only regularity assumption on the solutions (actually on the initial data) is that they have finite energy. The main results are: (1) the existence of dispersive solutions at -\(\infty\), in the sense that for any finite energy solution \(u_-\) of the free equation \(u_{tt}-\Delta u+m^ 2u=0\), there exists a global solution u of (*) such that \(u(t)- u_-(t)\to 0\) in the energy norm as \(t\to -\infty\). This implies the existence of the wave operator \(\Omega_-\) defined in the whole energy space. (Similar results hold of course at \(+\infty)\); (2) the fact that for small initial data, all solutions thereby obtained are dispersive both at \(+\infty\) and -\(\infty\), namely asymptotic completeness for small initial data. This paper complements earlier work of \textit{W. Strauss} [J. Funct. Anal. 41, 110-133 (1981; Zbl 0466.47006) and ibid. 43, 281-293 (1981; Zbl 0494.35068)] where the same results are proved (for any dimension \(n\geq 2)\) in the range 4/n\(\leq p-1\leq 4/(n-1)\) (i.e. 4/3\(\leq p- 1\leq 2\) for \(n=3)\). The method consists in solving the integral equation \[ u_ s(t)=u_-(t)+\sum^{t}_{s}d\tau E(t-\tau)h(u_ s(\tau)) \] where \(E(t)=\omega^{-1}\sin \omega t\), \(\omega =(-\Delta +m^ 2)^{1/2}\), and taking the limit of \(u_ s\) as \(s\to -\infty\) by a compactness method. The basic tools are (1) space time integrability properties of solutions of the free equation proved by Strichartz, and (2) Besov space estimates of E(t) due to Brenner and Pecher. Similar results, actually for \(2\leq n\leq 5\), have been obtained independently in [\textit{H. Pecher}, Math. Z. 185, 261-270 (1984; Zbl 0538.35063)]. They have been extended to arbitrary dimension in [\textit{H. Pecher}, Low energy scattering for non-linear Klein-Gordon equations, preprint (1984)] and [\textit{M. Tsutsumi} and \textit{N. Hayashi}, Scattering of solutions of non-linear Klein-Gordon equations in higher space dimensions. Nonlinear partial differential equations in applied science, Proceedings of the U.S.-Japan Seminar, Tokyo 1982, Vol. 6, North-Holland Math. Stud. (1983)]. The asymptotic completeness problem in the energy space without smallness condition has been treated in [\textit{P. Brenner}, On scattering and everywhere defined scattering operators for non-linear Klein-Gordon equations, J. Differ. Equations (to appear); Math. Z. 186, 383-391 (1984; Zbl 0524.35084)].
scientific article; zbMATH DE number 3882887