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Problème de Cauchy pour des systèmes hyperboliques semi-linéaires. (French) Zbl 0566.35068
The author considers the Cauchy problem for certain semilinear hyperbolic systems such as the Schrödinger-Klein-Gordon equations and the coupled Schrödinger equations of the type $$-i\psi_ t-\Delta \psi =F(\psi,\phi);\quad -i\phi_ t-\Delta \phi =G(\psi,\phi)$$ in three space dimensions. - In contrast to former results the nonlinearities are general enough to exclude the use of energy conservation but allow the use of charge conservation. Essentially they have to grow quadratically. Under these assumptions a global existence and uniqueness result is proven e.g. in the space $$C^ 0({\mathbb{R}},L^ 4({\mathbb{R}}^ 3)\cap L^ 2({\mathbb{R}}^ 3)).$$ The necessary a-priori-bounds can be given by use of the well-known $$L^ p-L^{p'}$$-estimates for the solutions of the linear problem. - In the second part of the paper the Dirac-Klein-Gordon system with a generalization of the Yukawa coupling is considered and a local existence and uniqueness result is proven.
Reviewer: H.Pecher

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35G25 Initial value problems for nonlinear higher-order PDEs
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