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The global Cauchy problem for the nonlinear Klein-Gordon equation. (English) Zbl 0549.35108
This paper is devoted to the proof of existence and uniqueness of the solutions of the Cauchy problem for the nonlinear Klein-Gordon (NLKG) equation \(\square\phi +f(\phi)=0\), with \(\phi\) a complex function defined in space time \({\mathbb{R}}^{n+1}\) and f a nonlinear interaction satisfying suitable power bounds at infinity, typically \(f(\phi)=(\lambda_ 1|\phi |^{p_ 1-1}+\lambda_ 2|\phi |^{p_ 2- 1})\phi,\) with \(1\leq p_ 1\leq p_ 2\) and \(\lambda_ 2>0\). The result is proved for arbitrary initial data of finite energy, namely \((\phi,{\dot\phi })\in H^ 1\oplus L^ 2,\) for arbitrary space dimension. The authors rely for the existence of (weak) solutions on the known results of Lions, Segal, Strauss and others, and merely recall them briefly in a form close to that given by J. L. Lions [Quelques méthodes de résolution des problèmes aux limites non linéaires (1969; Zbl 0189.406)], with minor additions. The original part of the paper is the treatment of the uniqueness problem. It combines two ingredients. The first one is a proof that all solutions of NLKG with finite energy satisfy local (in time) space time integrability properties similar to those known to hold for the free wave equation. That proof proceeds through estimates of the solutions in homogeneous Sobolev spaces. The second one is a partial contraction method of suitable \(L^ r\)-norms of the solutions on bounded subsets of the energy space, and uses estimates of Strichartz and Peral on the free wave equation. The paper ends with a brief appendix on homogeneous Sobolev spaces. The authors have previously treated the nonlinear Schrödinger equation with similar results [see: The global Cauchy problem for the nonlinear Schrödinger equation revisited (preprint)].

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
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