The scattering theory for the nonlinear wave equation with small data. II
[For part I see J. Math. Kyoto Univ. 25, 703-715 (1985; Zbl 0605.35069).] In Part I we studied a scattering problem for the nonlinear wave equation \[ (0.1)\quad \partial^ 2_ tw(t)=\Delta w(t)+f(w(t))=0\quad in\quad (x,t)\in {\mathbb{R}}^ n\times {\mathbb{R}}. \] Here \(n\geq 2\) and \(\Delta =\sum^{n}_{j=1}\partial^ 2_{x_ j}\). For purposes of illustration, let \(f(w)=\lambda | w|^{\rho -1}w\) (\(\lambda\in {\mathbb{R}}\), \(\rho >1)\). It was then proved that the scattering operator exists on a dense set of a neighborhood of 0 in the energy space if \(\rho\) is restricted to satisfy \((0.2)\quad \rho \geq 2\) and \[ (0.3)\quad \gamma (n)\equiv (n^ 2+3n-2+\sqrt{(n^ 2+3n-2)^ 2-8n(n-1)})/(2n(n- 1))<\rho \leq (n+3)/(n-1). \] The lower bound \(\gamma\) (n) of \(\rho\) is smaller than those obtained in \textit{W. A. Strauss} [J. Funct. Anal. 41, 110-133 (1981; Zbl 0466.47006)] and \textit{S. Klainerman} [Arch. Ration. Mech. Anal. 78, 73-98 (1982; Zbl 0502.35015)], and it is expected to give a critical power. On the other hand, (0.2) is too restrictive. For it contradicts (0.3) in case of high space dimension (n\(\geq 5)\). The main purpose of the present paper is to eliminate this restriction.
scientific article; zbMATH DE number 4082125