Myshkis, A. D.; Filimonov, A. M. On the global continuous solvability of the mixed problem for one-dimensional hyperbolic systems of quasilinear equations. (English. Russian original) Zbl 1152.35071 Differ. Equ. 44, No. 3, 413-427 (2008); translation from Differ. Uravn. 44, No. 3, 394-407 (2008). Summary: We consider a hyperbolic system of quasilinear equations written in Riemann invariants for the case of one spatial variable. For this system, we obtain sufficient conditions for the global generalized continuous solvability of the mixed problem in the class of functions monotone with respect to \(x\) for arbitrary \(t\) and with respect to \(t\) for \(x= 0\). In contrast to earlier studies, we assume that the boundary conditions may depend not only on time but also on the unknown functions. Cited in 2 Documents MSC: 35L50 Initial-boundary value problems for first-order hyperbolic systems 35L60 First-order nonlinear hyperbolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) Keywords:nonlinear boundary conditions; Riemann invariants; one spatial variable PDF BibTeX XML Cite \textit{A. D. Myshkis} and \textit{A. M. Filimonov}, Differ. Equ. 44, No. 3, 413--427 (2008; Zbl 1152.35071); translation from Differ. Uravn. 44, No. 3, 394--407 (2008) Full Text: DOI References: [1] Lax, P., J. Math. Phys., 1964, vol. 5, no. 5, pp. 611–613. · Zbl 0135.15101 · doi:10.1063/1.1704154 [2] Zabusky, N., J. Math. Phys., 1962, vol. 3, no. 5, pp. 1028–1099. · Zbl 0118.41802 · doi:10.1063/1.1724290 [3] Fermi, E., Pasta, J.R., and Ulam, S., Studies of Nonlinear Problems, Los Alamos Report no. 1940, May, 1955. [4] John, F., Comm. Pure Appl. Math., 1974, vol. 27, no. 2, pp. 377–405. · Zbl 0302.35064 · doi:10.1002/cpa.3160270307 [5] Myshkis, A.D. and Filimonov, A.M., Differ. Uravn., 1981, vol. 17, no. 3, pp. 488–500. [6] Hoff, D., J. Math. Anal. Appl., 1982, vol. 86, pp. 221–236. · Zbl 0488.35057 · doi:10.1016/0022-247X(82)90266-9 [7] Li Tatsien and Qin Tiehu, Chinese Ann. Math. Ser. B, 1985, vol. 6, no. 2, pp. 199–210. [8] Filimonov, A.M., Sufficient Conditions for the Global Solvability of the Mixed Problem for Quasilinear Hyperbolic Systems with Two Independent Variables, Deposited in VINITI, Moscow, 1981, no. 6-81. [9] Qin Tiehu, Chinese Ann. Math. Ser. B, 1985, vol. 6, no. 3, pp. 289–298. [10] Abolinya, V.E. and Myshkis, A.D., Mat. Sb., 1960, vol. 50, no. 4, pp. 423–442. [11] Ptashnik, B.I., Il’kiv, V.S., Kmit’, I.Ya., and Polishchuk, V.M., Nelokal’ni kraiovi zadachii dlya rivnyan’ iz chastinnimi pokhidnimi (Nonlocal Boundary Value Problems for Partial Differential Equations), Kiev, 2002. [12] Andrusyak, R.V. and Kirilich, V.M., Dop. Nats. Akad. Nauk Ukrainy, 2005, no. 7, pp. 7–11. [13] Turo, J., Ann. Polon. Math., 1991, vol. 52, pp. 231–238. · Zbl 0729.35139 · doi:10.4064/ap-52-3-231-238 [14] Turo, J., Nonlinear Anal., 1997, vol. 30, pp. 2329–2340. · Zbl 0893.35135 · doi:10.1016/S0362-546X(96)00162-9 [15] Sharkovskii, A.N., Maistrenko, Yu.L., and Romanenko, E.Yu., Raznostnye uravneniya i ikh prilozheniya (Difference Equations and Their Applications), Kiev: Naukova Dumka, 1986. · Zbl 0669.39001 [16] Myshkis, A.D. and Filimonov, A.M., in Nelineinyi analiz i nelineinye differentsial’nye uravneniya (Nonlinear Analysis and Nonlinear Differential Equations), Trenogina, V.A. and Filippova, A.F., Eds., Moscow: Fizmatlit, 2003, pp. 337–351. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.