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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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References:
[1] Bachelot, A., Problème de Cauchy pour des systèmes de klein‐gordon‐schroe‐dinger, C. R. Acad. Sc. Paris, t. 296, 525-528, (1983) · Zbl 0538.35016
[2] J. B. Baillon et J. M. Chadam, The Cauchy problem for the coupled Schroedinger‐Klein‐Gordon equations; in Contemporar Developments in Continuum Mechanics and Partial differential equations, G. M. de la Penha, L. A. Medeiros (Eds), North‐Holland Publishing Company, 1978. · Zbl 0404.35084
[3] M. Balabane, H. Emani‐Rad, Pseudodifferential parabolic systems in L^p(ℝ^n); in « Contriburion to nonlinear partial differential equations » C. Bardos, A. Damlamian, J. I. Diaz, J. Hernandez (Eds), Research Notes in Maths 89, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1983.
[4] Ph. Brennbr, The Cauchy problem for systems in L_p and L_p,α, Ark. Mat., t. 11, 75-101, (1973)
[5] Ph. Brenner, V. Thomee, L. B. WAHLBIN, Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Math., t. 434, Springer‐Verlag, Berlin, Heidelberg, New‐York, 1975. · Zbl 0294.35002
[6] Chadam, J., Global solutions of the Cauchy problem for the (classical) coupled maxwell‐dirac equations in one space dimension, J. Func. Anal., t. 13, 173-184, (1973) · Zbl 0264.35058
[7] Chadam, J., Asymptotic behaviour of equations arising in quantum field theory, J. Applic. Anal., t. 3, 377-402, (1973) · Zbl 0339.35059
[8] Chadam, J.; Glassey, R., On the maxwell‐dirac equations with zero magnetic field and their solutions in two space dimensions, J. Math. Anal. and Appl., t. 53, 495-507, (1976) · Zbl 0324.35076
[9] Chadam, J.; Glassey, R., On certain global solutions of the Cauchy problem for the (classical) coupled klein‐gordon‐dirac equations in one and three space dimensions, Arch. Rat. Mech. Anal., t. 54, 223-237, (1974) · Zbl 0285.35042
[10] Choquet‐Bruhat, Y., Solutions globales des équations de maxwell‐dirac‐klein‐gordon (masses nulles), C. R. Acad. Sc. Paris, t. 292, 153-158, (1981) · Zbl 0498.35053
[11] Da Prato, G.; Giusti, E., Equazioni di Schrödinger e delle onde per l’operatore di Laplace iterato in L^p(ℝ^n), Ann. Mat. Pura e Appl., t. 76, 377-397, (1967) · Zbl 0171.07702
[12] Fukuda, I.; Tsutsumi, M., On coupled klein‐gordon‐schrödinger equations, I, Bull. Sci. Engeg. Res. Lab. Waseda Univ., t. 69, 51-62, (1975)
[13] Fukuda, I.; Tsutsumi, M., On coupled klein‐gordon‐schrödinger equations, II, J. Math. Anal. and Applic., t. 66, 358-378, (1978) · Zbl 0396.35082
[14] Fukuda, I.; Tsutsumi, M., On coupled klein‐gordon‐schrödinger equations, III, Math. Japonica, t. 24, 3, 307-321, (1979) · Zbl 0415.35071
[15] Gross, L., The Cauchy problem for the coupled maxwell‐dirac equations, Comm. Pure and Appl. Math., t. 19, 1-15, (1966) · Zbl 0137.32401
[16] Holder, JR, E. J., On the existence, scattering, and blow up of solutions to systems of nonlinear Schrödinger equations, Indiana Univ. Math. J., t. 30, 653-673, (1981) · Zbl 0595.35034
[17] Hörmander, L., Estimates for translation invariant operators in L^p‐spaces, Acta Math., t. 104, 93-140, (1960) · Zbl 0093.11402
[18] Littman, W., The wave operator and L^p norms, J. Math. Mech., t. 12, 55-68, (1963) · Zbl 0127.31705
[19] Marshall, B.; Strauss, W.; Wainger, S., L^p-L^p′ estimates for the klein‐gordon equation, J. Math. Pures et Appl., t. 59, 417-440, (1980) · Zbl 0457.47040
[20] Peral, J., L^p estimates for the wave equation, J. Func. Anal., t. 36, 114-145, (1980) · Zbl 0442.35017
[21] M. Reed, Abstract nonlinear wave equations; Lecture Notes in Math., t. 507, Springer‐Verlag, 1976. · Zbl 0317.35002
[22] Reed, M.; Simon, B., Fourier analysis, self adjointness, Methods of modern mathematicalphysics, Vol. 2, (1975), Academic Press New York · Zbl 0308.47002
[23] Segal, J., Nonlinear semi‐groups, Aml. Math., t. 78, 339-364, (1963) · Zbl 0204.16004
[24] Sjöstrand, S., On the Riesz means of solutions of Schrödinger equation, Ann. Scuola Norm. Sup. Pisa, t. 24, 331-348, (1970) · Zbl 0201.14901
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