Abstract:
A Riemann representation is obtained for the solutions of the Cauchy problem for a first order hyperbolic system in two independent variables, and applications of this construction to the study of the stability and dichotomy of solutions are indicated.
The main results of this paper have been announced earlier (see RZh.Mat., 1983, 4B433).
Bibliography: 8 titles.
\Bibitem{Rom85}
\by R.~K.~Romanovskii
\paper On~Riemann matrices of the first and second kind
\jour Math. USSR-Sb.
\yr 1986
\vol 55
\issue 2
\pages 485--492
\mathnet{http://mi.mathnet.ru/eng/sm2010}
\crossref{https://doi.org/10.1070/SM1986v055n02ABEH003016}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=806513}
\zmath{https://zbmath.org/?q=an:0662.35067|0583.35075}
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https://www.mathnet.ru/eng/sm/v169/i4/p494
This publication is cited in the following 25 articles:
A. N. Mironov, E. F. Koskova, “O zadache s usloviyami na kharakteristikakh i svobodnoi poverkhnosti dlya giperbolicheskoi sistemy uravnenii s tremya nezavisimymi peremennymi s dvukratnymi kharakteristikami”, Izv. vuzov. Matem., 2025, no. 1, 28–36
A. N. Mironov, A. P. Volkov, “On the Darboux problem for a hyperbolic system of equations with multiple characteristics”, Russian Math. (Iz. VUZ), 66:8 (2022), 31–36
A. N. Mironov, L. B. Mironova, Yu. O. Yakovleva, “Metod Rimana dlya uravnenii s dominiruyuschei chastnoi proizvodnoi (obzor)”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 25:2 (2021), 207–240
A. N. Mironov, L. B. Mironova, “Riemann–Hadamard Method for One System in Three-Dimensional Space”, Diff Equat, 57:8 (2021), 1034
L. B. Mironova, “Application of Riemann method to one system in three-dimensional space”, Russian Math. (Iz. VUZ), 63:6 (2019), 42–50
R. K. Romanovskii, Yu. A. Medvedev, “Boundary control of a hyperbolic system with one space variable”, Diff Equat, 53:3 (2017), 345
R. K. Romanovskii, Yu. A. Medvedev, “Optimal two-sided boundary control of heat transmission in a rod. Hyperbolic model”, Russian Math. (Iz. VUZ), 60:6 (2016), 45–51
Romanovskii R.K., Churasheva N.G., “Optimal Boundary Control of Heat Transfer in a One-Dimensional Material: a Hyperbolic Model”, Differ. Equ., 48:9 (2012), 1236–1244
Mamchuev M.O., “Cauchy Problem in Nonlocal Statement for a System of Fractional Partial Differential Equations”, Differ. Equ., 48:3 (2012), 354–361
R. K. Romanovskii, T. V. Ivanchenko, “Giperbolicheskaya model diffuzii neitronov v odnomernom zamedlitele”, Sib. zhurn. industr. matem., 14:4 (2011), 76–85
Romanovskii R.K., Sheblov A.V., “On the Riemann Matrices of the Hyperbolic System Dual to the System of Equations of One-Dimensional Gas Dynamics”, Differ. Equ., 47:6 (2011), 832–840
Romanovskii R.K. Bel'gart L.V., “On the Exponential Dichotomy of Solutions of the Cauchy Problem for a Hyperbolic System on a Plane”, Differ. Equ., 46:8 (2010), 1135–1144
Romanovskii R.K., Stratilatova E.N., Sheblov A.V., “O vliyanii malykh nachalnykh vozmuschenii na resheniya kvazilineinoi giperbolicheskoi sistemy”, Dokl. Akademii nauk vysshei shkoly Rossiiskoi Federatsii, 2009, no. 2, 42–50
R. K. Romanovskii, O. G. Zhukova, “Granichnoe upravlenie protsessom teploperenosa v dvumernom materiale. Giperbolicheskaya model”, Sib. zhurn. industr. matem., 11:3 (2008), 119–125
Mendziv M.V. Romanovskii R.K., “Direct Lyapunov Method for Hyperbolic Systems on the Plane with Time-Periodic Coefficients”, Differ. Equ., 44:2 (2008), 267–273
Zhukova O.G., “Boundary Control of a Hyperbolic System of Heat Equations”, Differ. Equ., 44:1 (2008), 85–91
O. G. Zhukova, R. K. Romanovskii, “Dvustoronnee granichnoe upravlenie protsessom teploperenosa v odnomernom materiale. Giperbolicheskaya model”, Sib. zhurn. industr. matem., 10:4 (2007), 32–40
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