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Remarks on the Klein-Gordon equation. (English) Zbl 0655.35057

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1987, Exp. No. 1, 9 p. (1987).
From the work of S. Klainerman [Commun. Pure Appl. Math. 38, 631- 641 (1985; Zbl 0597.35100)] and J. Shatah [ibid. 38, 685-696 (1985; Zbl 0597.35101)] it is known that for nonlinear perturbations of the Klein-Gordon equation in \({\mathbb{R}}^{1+n}\), \[ (1.1)\quad \square u+u=F(u,u',u''),\quad \square =\partial^ 2_ 0-\partial^ 2_ 1-...- \partial^ 2_ n, \] F vanishes of second order at 0, and F is linear in u”, the Cauchy problem with small data in \(C^{\infty}_ 0\) has a global solution if \(n\geq 3.\)
The main purpose of this paper is to examine the remaining cases \(n=1,2\). We begin by studying in Section 2 the solutions of the unperturbed Klein- Gordon equation \(\square u+u=0\) in considerable detail for arbitrary n. This covers the estimates of W. von Wahl [Math. Z. 120, 93-106 (1971; Zbl 0212.442)] and gives in addition a much more precise description of the asymptotic properties to serve as a goal in the study of (1.1). In Section 3 we discuss \(L^ 2\) estimates for the inhomogeneous linear Klein-Gordon equation in the spirit of Klainerman (loc. cit.). His estimates for the case \(n=3\) were not sharp but sufficient to establish global existence theorems then. Their analogue for \(n=1\) or \(n=2\) would not give a good estimate for the lifespan of the solutions. We therefore reexamine the estimates of Klainerman (loc. cit.) for arbitrary dimension, but some of them may not be sharp when \(n>3\). Using these bounds we outline in Section 4 how existence theorems for (1.1) follow when F vanishes of second order or of third order at 0. In the second case we believe that our results are optimal, but it is feasible that the lifespan of the solutions must be of the same order of magnitude in the two cases. Some evidence in favor of that is represented in Section 5. In particular we discuss the case \(n=0\) there, that is, the ordinary differential equation \(u''+u=F(u,u').\)
Some new idea seems needed to decide what the optimal results should be when \(n=1\) or \(n=2\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B20 Perturbations in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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