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Mixed problems for quasilinear hyperbolic systems. (English) Zbl 0893.35135
The paper deals with the quasilinear hyperbolic system of partial differential functional equations in diagonal form $D_xz_i+ \lambda_i(x,y,z,Vz) D_yz_i= f_i(x,y, z,Vz), \quad i\in\{1, \dots,n\}, \tag{1}$ where $$z(x,y)= (z_1(x,y), \dots, z_n(x,y))$$ and $$V$$ is an operator of Volterra type. The author studies local generalized solutions of the system (1) with mixed conditions, continuous dependence of the generalized solutions on the given functions as well as existence of global generalized solutions.
Reviewer: D.Bainov (Sofia)

##### MSC:
 35R10 Functional partial differential equations 35D05 Existence of generalized solutions of PDE (MSC2000)
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##### References:
 [1] Abolinia, V.E.; Myshkis, A.D., Mixed problem for semilinear hyperbolic system on the plane (in Russian), Mat. sb., 50, 4, 423-442, (1960) · Zbl 0152.30001 [2] Bassanini, P.; Turo, J., Generalized solutions to free boundary problems for hyperbolic systems of functional partial differential equations, Ann. mat. pura appl., 156, 211 230, (1990) · Zbl 0716.35088 [3] Doktor, A., Global solution of mixed problem for a certain system of nonlinear conservation laws, Czech. math. J., 27, 69-95, (1977) · Zbl 0347.35056 [4] Filimonov, A.M., Local solutions of mixed problem for hyperbolic system of quasilinear equations in two variables (in Russian), (1982), (preprint) [5] Filimonov, A.M., Sufficient conditions of global solvability of a mixed problem for quasilinear hyperbolic systems in two independent variables (in Russian), (1980), (preprint) [6] Johanson, J.L.; Smoller, J.A., Global solutions for certain systems of quasilinear hyperbolic equations, J. math. mech., 17, 561-572, (1967) [7] rilicz, W.M.; Myshkis, A.D., Generalized semilinear hyperbolic Stefan problem on the straight line, Diff: urav., 27, 497-503, (1991) [8] Kozakov, K.Y.; Morozov, S.F., On definiteness of an unknown discontinuity line of a solution of mixed problems for a quasilinear hyperbolic system (in Russian), Ukr. mat. zum., 37, 443-450, (1985) [9] Myshkis, A.D.; Filimonov, A.M., Continuous solutions of quasilinear hyperbolic systems in two independent variables (in Russian), (), 524-529, ulgaria · Zbl 0549.35076 [10] Myshkis, A.D.; Filimonov, A.M., Continuous solutions of quasilinear hyperbolic systems in two independent variables (in Russian), Diff. urav., 17, 488-500, (1981) · Zbl 0459.35052 [11] Turo, J., Generalized solutions of mixed problems for quasilinear hyperbolic systems of functional partial differ-ential equations in the Schauder canonic form, Ann. polon. math., 50, 157-183, (1989) · Zbl 0717.35051 [12] Turo, J., Local generalized solutions of mixed problems for quasilinear hyperbolic systems of functional partial differential equations in two independent variables, Ann. polon. math., 49, 259-278, (1989) · Zbl 0685.35065 [13] Turo, J., Generalized solutions to functional partial differential equations of the first order, Zeszyty naukowe politechniki gdańskiej, matematyka, 14, 1-99, (1988) [14] Turo, J., Global solvability of the mixed problems for first order functional partial differential equations, Ann. polon. math., 52, 231-238, (1991) · Zbl 0729.35139
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