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On some class of quasilinear hyperbolic systems of partial differential functional equations of the first order. (English) Zbl 0612.35082
Etant donné le système hyperbolique quasi-linéaire \[ \sum^{n}_{\partial =1}A_{ij}(x,y,z)[\frac{\partial z_ j(x,y)}{\partial x}+\sum^{m}_{k=1}\rho_{ik}(x,y,z,v(z))\frac{\partial z_ j}{\partial y_ k}]=f_ i(x,y,z,v(z)) \] z(x,y)\(=\gamma (x,y)\) \((x,y)\in [-2,0]\times {\mathbb{R}}^ m\), l’auteur met en évidence un théorème d’existence et d’unicité (§ 5) de solutions généralisés du problème.
Pour cela il introduit un ensemble J de fonctions \(\gamma\), puis à tout \(\gamma\) de J un ensemble \(K_{\gamma}\), ainsi qu’une application F, dont il montre, après un bon choix d’hypothèses, qu’elle est une contraction stricte de \(K_{\gamma}\) dans lui-même, ce qui entrainera l’existence d’une unique solution du problème dans la classe \(K_{\gamma}.\)
Le résultat s’applique, en particulier, à des systèmes hyperboliques avec retard, ainsi qu’à des systèmes d’équations intégro-différentielles.
Reviewer: M.Lacroix

MSC:
35L60 First-order nonlinear hyperbolic equations
35R10 Functional partial differential equations
45K05 Integro-partial differential equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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References:
[1] P. Bassanini: Su una recente dimostrazione circa il problema di Cauchy per sistemi quasi lineari iperbolici. Boll Un. Mat. Ital. (5) 13-B (1976), 322-335. · Zbl 0342.35035
[2] P. Bassanini: Iterative methods for quasilinear hyperbolic systems. Boll. Un. Mat. Ital. (6) 1-B (1982), 225-250. · Zbl 0488.35056
[3] P. Bassanini: The problem of Graffi-Cesari. Proceedings of Inter. Conference on Nonlinear Phenomena in Math. Sci., Arlington, USA, 1980, Acad. Press 1982, 87-101.
[4] P. Bassanini M. C. Salvatori: Un problema ai limiti per sistemi integrodifferenziali non lineari di tipo iperbolico. Boll. Un. Mat. Ital. (5) 18-B (1981), 785-798. · Zbl 0492.45010
[5] G. Caginalp: Nonlinear equations and systems in several space variables. Journal of Diff. Equations 48 (1983), 71 - 94. · Zbl 0513.35050 · doi:10.1016/0022-0396(83)90060-8
[6] G. Caginalp: Nonlinear equations with coefficients of bounded variation in two space variables. Journal of Diff. Equations 43 (1982), 134-155. · Zbl 0451.35018 · doi:10.1016/0022-0396(82)90078-X
[7] L. Cesari: A boundary value problem for quasilinear hyperbolic systems in the Schauder canonic form. Ann. Scuola Norm. Sup. Pisa, (4) 1 (1974), 311 - 358. · Zbl 0307.35063 · numdam:ASNSP_1974_4_1_3-4_311_0 · eudml:83681
[8] L. Cesari: A boundary value problem for quasilinear hyperbolic systems. Riv. Mat. Univ. Parma, 3 (1974), 107-131. · Zbl 0342.35036
[9] M. Cinquini-Cibrario: Teoremi di esistenza per sistemi di equazioni quasilineari a derivate parziali in piu variabili independenti. Ann. Mat. Pura Appl., 75 (1967), 1 - 46. · Zbl 0158.11301 · doi:10.1007/BF02416798
[10] A. Doktor: Global solution of mixed problem for certain system of nonlinear conservation laws. Czech. Math. J. 27 (102), (1977), 69-95. · Zbl 0347.35056 · eudml:12983
[11] E. Hopf: The partial differential equation \(u_t + uu_x = \mu u_{xx}\). Comm. Pure Appl. Math, 3 (1950), 201-230. · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[12] Z. Kamont: Existence of solutions of first order partial differential-functional equations. Ann. Soc. Math. Polon., Ser. I: Comm. Math. · Zbl 0726.35134
[13] Z. Kamont J. Turo: On the Cauchy problem for quasilinear hyperbolic system of partial differential equations with a retarded argument. Boll. Uh. Mat. Ital. · Zbl 0614.35089
[14] С. Н. Кружков: Квазилинейные уравнения первого порядка со многими независимыми переменными. Математический Сборник 81 (1970), 230-255. · Zbl 1170.92319 · doi:10.1016/0022-5193(70)90090-1
[15] С. Н. Крулсков: Обобщенные решения нелинейных уравнений первого порядка со многими независимыми переменными. Мат. Сборник 70 (1966), 394-415. · Zbl 1155.78304 · doi:10.1109/TAP.1966.1138693
[16] О. А. Олейник: Разрывные решения нелинейных дифференциальных уравнений. Ncnexn Mat. Hauk 12 (75), (1977), 3-73. · Zbl 1170.01341
[17] Б. Л. Рожрественский Н. Н. Яненко: Системы квазилинейных уравнений. Москва 1978. · Zbl 1234.93001
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