Far Eastern Mathematical Journal

To content of the issue


Boundary value problem for third order equation with multiple characteristics and alternating function on the highest derivative


Kozhanov A.I., Potapova S.V.

2017, issue 1, Ñ. 48-58


Abstract
In this paper we investigated the regular solvability of conjugate problem (generalized diffraction problem) for third order equation with multiple characteristics and alternating function on the highest derivative. This function has a discontinuity of the first kind and changes sign when passing the point of discontinuity. The existence and uniqueness of regular solutions are proved by the regularization and continuation methods.

Keywords:
equations with multiple characteristics, equations with changing time direction, discontinuous coefficients, conjugate problem, regular solutions, existence and uniqueness

Download the article (PDF-file)

References

[1] L. Cattabriga, Annali della scuola normale Superiore di pisa e mat, 13:2, (1956), 163–203.
[2] L. Cattabriga, “Potenziali di linea e di dominio per equazioni non paraboliche in due variabili a characteristiche multiple”, Rend. Semin. Mat. Univ. Padova, 3, (1961), 1–45.
[3] T.D. Dzhuraev, Kraevye zadachi dlia uravneniia smeshannogo i smeshanno-sostavnogo tipov, FAN, Tashkent, 1986.
[4] S. Abdinazarov, “Obshchie kraevye zadachi dlia uravneniia tret'ego poriadka s kratnymi kharakteristikami”, Differentsial'nye uravneniia, 17, (1981), 3–12.
[5] M. Mascarello, L. Rodino, Partial di?erentional equations with multiple characteristics, Wiley, Berlin, 1997.
[6] M. Mascarello, L. Rodino, M. Tri, “Partial di?erentional operators with multiple symplectic characteristics”, Partial di?erential equations and spectral theory, ed. M. Demuth, B.-W. Schulze, Birkhauser, Basel, 2001, 293–297.
[7] L. Rodino, A. Oliaro, “Solvability for semilinear PDE with multiple characteristics”, Evolution equations, v. 60, ed. R. Picard, M. Reissig, W. Zajaczkowski, Banach Center Publ., Warsaw, 2003, 295–303.
[8] A.I. Kozhanov, “O razreshimosti nelokal'noi po vremeni zadachi dlia odnogo uravneniia s kratnymi kharakteristikami”, Mat. zametki IaGU, 8:2, (2001), 27–40.
[9] A.I. Kozhanov, “Composite Type Equations and Inverse Problems”, Utrecht, the Netherlands, VSP, 1999.
[10] A. R. Khashimov, A.M. Turginov, “O nekotorykh nelokal'nykh zadachakh dlia uravneniia tret'ego poriadka s kratnymi kharakteristikami”, Mat. zametki SVFU, 21:1, (2014), 69–74.
[11] G.G. Doronin, N.A. Larkin, E. Tronco, “Exponential Decay of Weak Solutions for the Zakharov-Kuznetsov Equation”, Nonclassical equations of mathematical physics. 1ed. Novosibirsk, 446, (2012), 5–13.
[12] A.M. Abdrakhmanov, A.I. Kozhanov, “Zadacha s nelokal'nym granichnym usloviem dlia odnogo klassa uravnenii nechetnogo poriadka”, Izvestiia vuzov. Matematika, 5, (2007), 3–12.
[13] N.A. Larkin, “Korteweg–de Vries and Kuramoto–Sivashinsky equations in bounded domains”, J. Math. Anal. Appl., 297:2, (2004), 169–185.
[14] B.A. Bubnov, “Generalized boundary value problems for Korteweg–de Vries equation in bounded domains”, Di?erential Equations, 15, (1979), 17–21.
[15] B.B. Khablov, “O nekotorykh korrektnykh postanovkakh granichnykh zadach dlia uravneniia Kortevega de Friza, Preprint In-ta matem. SO AN SSSR, Novosibirsk, 1979.
[16] A.V. Faminskii, N.A. Larkin, “Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval”, Electronic Journal of Di?. Equations, 2010, 1–20.
[17] A.V. Faminskii, N.A. Larkin, “Odd-order quasilinear evolution equations posed on a bounded interval”, Bol. Soc. Paran. Mat., 28:1, (2010), 67–77.
[18] Sh. Cui, Sh. Tao, “Strichartz estimates for dispersive equations and solvability of Kawahara equation”, J. Math. Anal. Appl., 304, (2005), 683–702.
[19] N.A. Larkin, “Correct initial boundary value problems for dispersive equations”, J. Math. Anal. Appl., 344:2, (2008), 079–1092.
[20] S. Abdinazarov, A. Khashimov, “Kraevye zadachi dlia uravneniia s kratnymi kharakteristikami i razryvnymi koeffitsientami”, Uz. mat. zhurn., 1993, ¹ 1, 3–12.
[21] A. Khashimov, “Ob odnoi zadache dlia uravneniia smeshannogo tipa s kratnymi kharakteristikami”, Uz. mat. zhurn., 1995, ¹ 2, 95–97.
[22] V.I. Antipin, “Razreshimost' kraevoi zadachi dlia uravneniia tret'ego poriadka s meniaiushchimsia napravleniem vremeni”, Matematicheskie zametki IaGU, 18:1, (2011), 8–15.
[23] V.I. Antipin, “Razreshimost' kraevoi zadachi dlia operatorno-differentsial'nykh uravnenii smeshannogo tipa”, Sibirskii matematicheskii zhurnal, 54:2, (2013), 245–257.
[24] S.G. Pyatkov, S. Popov, V.I. Antipin, “On solvability of boundary value problem for kinetic operator-di?erential equations”, Integral Equation and Operator Theory, 80:4, (2014), 557–580.
[25] M. Gevrey, “Sur les equations aux derivees partielles du type parabolique”, J. Math. Appl., 9:6, (1913), 305–478.
[26] N.A. Lar'kin, V.A. Novikov, N.N. Ianenko, Nelineinye uravneniia peremennogo tipa, Nauka, Novosibirsk, 1983.
[27] S.A. Tersenov, Parabolicheskie uravneniia s meniaiushchimsia napravleniem vremeni, Nauka, Novosibirsk, 1985.
[28] I.E. Egorov, S.G. Piatkov, S. V. Popov, Neklassicheskie differentsial'no–operatornye uravneniia, Nauka, Novosibirsk, 2000.
[29] N.V. Kislov, I.S. Pul'kin, “O sushchestvovanii i edinstvennosti slabogo resheniia zadachi Zhevre s obobshchennymi usloviiami skleiki”, Vestnik MEI, 2002, ¹6, 88–92.
[30] I.M. Petrushko, E.V. Chernykh, “O parabolicheskikh uravneniiakh 2-go poriad+ka s meniaiushchimsia napravleniem vremeni” , Vestnik MEI, 2003, ¹6, 85–93.
[31] R. Beals, “On an equations of mixed type from electron scattering”, J. Math. Anal. Appl., 568:1, (1977), 32–45.
[32] C.E. Siewert and P.E. Zweifel, “Radiative transfer, II”, J. Math. Phys., 7, (1966), 2092–2102.
[33] V.A. Trenogin, Funktsional'nyi analiz, Nauka, M., 1980.
[34] S.V. Potapova, “Boundary value problems for pseudohyperbolic equations with a variable time direction”, TWMS Journal of Pure and Applied Mathematics, 3:1, (2012), 75–91.

To content of the issue