Boundary-value problems for fourth-order equations of hyperbolic and composite types

V. I. Korzyuk ; O. A. Konopel'ko ; E. S. Cheb

Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 2, Tome 36 (2010), pp. 87-111.

Résumé

Boundary-value problems for fourth-order linear partial differential equations of hyperbolic and composite types are studied. The method of energy inequalities and averaging operators with variable step is used to prove existence and uniqueness theorems for strong solutions. The Riesz theorem on the representation of linear continuous functionals in Hilbert spaces is used to prove the existence and uniqueness theorems for generalized solutions.

Bibliographie

[1] Aldashev S. A., “O zadachakh dlya odnogo klassa mnogomernykh integrodifferentsialnykh giperbolicheskikh uravnenii”, Tretii sibirskii kongress po prikladnoi i industrialnoi matematike, posvyaschennyi pamyati S. L. Soboleva (1908–1989), INPRIM-98. Tezisy dokladov. ch. 4 (Novosibirsk, 1998), Izd-vo IM SO RAN, Novosibirsk, 1998, 137–140

[2] Akhiev S. S., Sanieva S. N., DAN SSSR, 316:3 (1991), 524–528 | MR | Zbl

[3] Berkhin P. E., “Ob odnoi kraevoi zadache dlya uravneniya sostavnogo tipa”, DAN SSSR, 24:3 (1974), 496–498

[4] Brish N. I., “Klassicheskie resheniya smeshannykh zadach dlya nestatsionarnykh uravnenii”, Differents. uravn., 1:4 (1965), 523–528 | MR | Zbl

[5] Brish N. I., Differents. uravn., 5:12 (1969), 2200–2215 | MR | Zbl

[6] Burenkov V. I., “O plotnosti beskonechno differentsiruemykh funktsii v prostranstvakh Soboleva dlya proizvolnogo otkrytogo mnozhestva”, Tr. MIAN SSSR, 131, 1974, 39–50 | MR | Zbl

[7] Burenkov V. I., “Priblizhenie beskonechno differentsiruemymi funktsiyami s sokhraneniem granichnykh znachenii”, Tr. MIAN SSSR, 180, 1987, 68–70 | Zbl

[8] Volevich L. R., Gindikin S. G., Smeshannaya zadacha dlya differentsialnykh uravnenii v chastnykh proizvodnykh s kvaziodnorodnoi starshei chastyu, Editorial URSS, M., 1999

[9] Vragov V. N., “Ob odnom uravnenii smeshanno-sostavnogo tipa”, Differents. uravn., 9:1 (1973), 132–134 | MR

[10] Gording L., Zadacha Koshi dlya giperbolicheskikh uravnenii, M., 1961 | Zbl

[11] Dainyak V. V., Korzyuk V. I., “Zadacha tipa Dirikhle dlya lineinogo differentsialnogo uravneniya tretego poryadka”, Differents. uravn., 23:5 (1987), 867–872 | MR

[12] Dzhuraev A., Sistemy uravnenii sostavnogo tipa, Nauka, M., 1972 | MR | Zbl

[13] Dzhuraev T. D., “Ob odnoi kraevoi zadache dlya uravneniya sostavnogo tipa”, Dokl. AN UzSSR, 1962, no. 4, 5–8 | Zbl

[14] Dzhuraev T. D., “Ob odnoi kraevoi zadache dlya uravneniya $\frac\partial{\partial x}\Delta u=0$”, Issledovaniya po differentsialnym uravneniyam, Tashkent, 1963, 46–51 | Zbl

[15] Dzhuraev T. D., “O edinstvennosti i suschestvovanii reshenii nekotorykh kraevykh zadach dlya uravnenii sostavnogo tipa”, Izv. AN UzSSR. Ser. fiz.-mat. nauk, 1967, no. 3

[16] Dzhuraev T. D., “O edinstvennosti resheniya nekotorykh kraevykh zadach dlya odnogo klassa uravnenii sostavnogo tipa”, Izv. AN UzSSR. Ser. fiz.-mat. nauk, 1967, no. 4

[17] Dzhuraev T. D., Rakhmanov U. O., “O korrektnykh kraevykh zadachakh dlya uravnenii smeshanno-sostavnogo tipa”, Differents. uravn., 9:1 (1973), 32–40 | Zbl

[18] Dzhuraev T. D., Sharifbaev Ya. S., “O nekotorykh kraevykh zadachakh dlya uravnenii s chastnymi proizvodnymi tretego poryadka”, Kraevye zadachi dlya differentsialnykh uravnenii, Fan, Tashkent, 1972

[19] Kozhanov A. I., “Kraevaya zadacha dlya odnogo klassa uravnenii tretego poryadka”, DAN SSSR, 249:3 (1979), 536–540 | MR | Zbl

[20] Kozhanov A. I., “Smeshannaya zadacha dlya odnogo klassa uravnenii neklassicheskogo tipa”, Differents. uravn., 15:2 (1979), 272–280 | MR | Zbl

[21] Kozhanov A. I., O razreshimosti kraevoi zadachi i svoistvakh reshenii dlya odnogo klassa uravnenii tretego poryadka, Preprint No 11, IM SO AN SSSR, 1986

[22] Konopelko O. A., “Granichnaya zadacha dlya uravneniya chetvertogo poryadka sostavnogo tipa”, Vestn. Belorusskogo un-ta. Ser. 1, 2008, no. 3, 73–79 | MR

[23] Korzyuk V. I., “Smeshannaya zadacha dlya nekotorykh nestatsionarnykh uravnenii s razryvnymi koeffitsientami”, Differents. uravn., 6:2 (1970), 343–357 | MR | Zbl

[24] Korzyuk V. I., “Energeticheskoe neravenstvo dlya kraevoi zadachi dlya giperbolicheskogo uravneniya s volnovym operatorom tretego poryadka”, Differents. uravn., 27:6 (1991), 1014–1022 | MR | Zbl

[25] Korzyuk V. I., “Ob operatorakh osredneniya s peremennym shagom. II”, Vestn. Belorusskogo un-ta. Ser. 1, 1992, no. 3, 63–65 | MR | Zbl

[26] Korzyuk V. I., “Pervaya smeshannaya zadacha dlya lineinogo giperbolicheskogo uravneniya vtorogo poryadka s odnorodnymi usloviyami na netsilindricheskoi oblasti”, Differents. uravn., 28:5 (1992), 847–856 | MR | Zbl

[27] Korzyuk V. I., “Metod energeticheskikh neravenstvo i operatorov osredneniya”, Vestn. Belorusskogo un-ta. Ser. 1, 1996, no. 2, 55–71 | MR | Zbl

[28] Korzyuk V. I., “Granichnaya zadacha dlya uravneniya Manzherona tretego poryadka”, Differents. uravn., 33:12 (1997), 1683–1694 | MR | Zbl

[29] Korzyuk V. I., “Kraevaya zadacha dlya giperbolicheskogo uravneniya s volnovym operatorom tretego poryadka”, Differents. uravn., 40:2 (2004), 208–215 | MR | Zbl

[30] Korzyuk V. I., Uravneniya matematicheskoi fiziki: kurs lektsii, v 6 ch., ch. 4, BGU, Minsk, 2008

[31] Korzyuk V. I., Dainyak V. V., “O slabom reshenii zadachi tipa Dirikhle dlya lineinogo differentsialnogo uravneniya tretego poryadka”, Differents. uravn., 28:6 (1992), 1056–1066 | MR | Zbl

[32] Korzyuk V. I., Dainyak V. V., “O razreshimosti smeshannykh zadach dlya nestatsionarnykh uravnenii tretego poryadka”, Vestn. Belorusskogo un-ta. Ser. 1, 2005, no. 3, 54–60 | MR | Zbl

[33] Korzyuk V. I., Konopelko O. A., “Obobschennye resheniya granichnykh zadach v tsilindricheskikh oblastyakh dlya uravneniya chetvertogo poryadka sostavnogo tipa”, X Belorusskaya matematicheskaya konferentsiya, Tez. dokl. Mezhdunar. nauch. konf. (Minsk, 3–7 noyabrya 2008 g.), ch. 4, Institut matematiki NAN Belarusi, Minsk, 2008, 69–71

[34] Korzyuk V. I., Cheb E. S., “Smeshannaya zadacha dlya giperbolicheskogo uravneniya chetvertogo poryadka”, Vestsi NAN Belarusi. Seryya fiz.-mat. n., 2004, no. 2, 9–13 | MR

[35] Korzyuk V. I., Cheb E. S., “Smeshannye zadachi dlya bivolnovogo uravneniya”, Vestn. Belorusskogo un-ta. Ser. 1, 2005, no. 1, 63–68 | MR

[36] Korzyuk V. I., Cheb E. S., “Granichnye zadachi dlya ellipticheskikh uravnenii vtorogo poryadka”, Tr. In-ta matematiki, 15, no. 2, 2007, 38–47

[37] Korzyuk V. I., Cheb E. S., “Zadacha Koshi dlya uravneniya chetvërtogo poryadka s bivolnovym operatorom”, Differents. uravn., 43:5 (2007), 669–676 | MR | Zbl

[38] Korzyuk V. I., Cheb E. S., Konopelko O. A., “O granichnykh zadachakh dlya uravnenii chetvertogo poryadka giperbolicheskogo i sostavnogo tipov”, Abstracts of the Fifth Int. Conf. on Differential and Functional Differential Equations (Moscow, Russia, August 17–24, 2008), 102–103

[39] Ladyzhenskaya O. A., “O metode Fure dlya volnovogo uravneniya”, DAN SSSR, 76:6 (1950), 765–768

[40] Ladyzhenskaya O. A., Smeshannaya zadacha dlya giperbolicheskogo uravneniya, Gostekhizdat, M., 1953

[41] Ladyzhenskaya O. A., “O razreshimosti osnovnykh kraevykh zadach dlya uravnenii parabolicheskogo i giperbolicheskogo tipov”, DAN SSSR, 97:3 (1954), 395–398 | Zbl

[42] Ladyzhenskaya O. A., “O razreshimosti osnovnykh kraevykh zadach dlya uravnenii parabolicheskogo i giperbolicheskogo tipov”, Mat. sb., 39(81):4 (1956), 491–524 | MR | Zbl

[43] Ladyzhenskaya O. A., “O nestatsionarnykh operatornykh uravneniyakh i ikh prilozheniyakh k lineinym zadacham matematicheskoi fiziki”, Mat. sb., 45(87):2 (1958), 123–158 | MR | Zbl

[44] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[45] Larkin N. A., “Kraevye zadachi v tselom dlya odnogo klassa giperbolicheskikh uravnenii”, Sib. mat. zhurn., 18:6 (1977), 1414–1419 | MR

[46] Larkin N. A., “Ob odnom klasse giperbolicheskikh vyrozhdayuschikhsya uravnenii”, Dinamika sploshnoi sredy, 36, 1978, 71–77 | MR

[47] Larkin N. A., “Ob odnom klasse kvazilineinykh giperbolicheskikh uravnenii, imeyuschikh resheniya v tselom”, DAN SSSR, 244:1 (1979), 38–41 | MR

[48] Levitan B. M., Mamatov A. E., “Otsenka matritsy-funktsii Koshi dlya sistemy Diraka v sluchae konechnozonnykh neperiodicheskikh potentsialov”, Mat. zametki, 53:4 (1993), 62–76 | MR | Zbl

[49] Lere Zh., Giperbolicheskie differentsialnye uravneniya, Mir, M., 1984 | MR

[50] Pyatkov S. V., “Ob odnom uravnenii sostavnogo tipa”, Differents. uravn., 16:1 (1980), 117–123 | MR | Zbl

[51] Salakhitdinov M. S., “O nekotorykh korrektnykh zadachakh dlya uravneniya”, DAN SSSR, 150:3 (1963) | Zbl

[52] Salakhitdinov M. S., “O teorii uravnenii sostavnogo tipa”, Izv. AN UzSSR, 1965, no. 3

[53] Salakhitdinov M. S., Uravneniya smeshanno-sostavnogo tipa, Fan, Tashkent, 1974 | MR

[54] Smirnov M. M., “Nekotorye kraevye zadachi dlya odnogo uravneniya smeshanno-sostavnogo tipa”, Sib. mat. zhurn., 5:4 (1964), 923–928 | Zbl

[55] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, L., 1950

[56] Terekhov A. N., O nekotorykh zadachakh dlya nelineinogo uravneniya sostavnogo tipa, Preprint, IM SO AN SSSR, Novosibirsk, 1980

[57] Uizem Dzh. B., Lineinye i nelineinye volny, Mir, M., 1977 | MR

[58] Eskin G. I., “Kraevaya zadacha dlya uravneniya $\frac\partial{\partial t}P(\frac\partial{\partial t},\frac\partial{\partial x})u=0$, $P(\frac\partial{\partial t},\frac\partial{\partial x})$ – ellipticheskii operator”, Sib. mat. zhurn., 3:6 (1962), 882–911 | MR | Zbl

[59] Eskin G. I., “Kraevye zadachi dlya uravnenii s postoyannymi koeffitsientami na ploskosti”, Mat. sb., 59(101) (1962), 67–104 | MR | Zbl

[60] Yurchuk N. I., Radyno Ya. V., Differents. uravn., 12:2 (1976), 331–342 | MR | Zbl

[61] Barozzi G. C., “Su alcuni problemi relativi a una equazione lineare a parziali di tipo composito”, Atti Semin. Mat. Fis. Univ. Modena, 10 (1961), 11–30 | MR | Zbl

[62] Bennish Joseph, “Mixed initial-boundary value problems for hyperbolic equations with constant coefficients”, J. Math. Anal. Appl., 153:2 (1990), 506–532 | DOI | MR | Zbl

[63] Burenkov V. I., Sobolev Spaces on Domains, Teubner, Stuttgart, 1998 | MR | Zbl

[64] Cattabriga L., “Su alcuni problemi per equazioni differenziali di tipo composito”, Rend. Sem. Mat. Univ. Padova, 27 (1957), 122–143 | MR | Zbl

[65] Davis R. B., “A boundary value problem for third-order linear partial differential equations of composite type”, Proc. Amer. Math. Soc., 3 (1952), 751–756 | DOI | MR | Zbl

[66] Eskin G., “Initial-boundary value problem for second order hyperbolic equations with general boundary conditions. I”, J. d'Analyse Math., 40 (1981), 43–89 | DOI | MR | Zbl

[67] Eskin G., “Initial-boundary value problem for second order hyperbolic equations with general boundary conditions. II”, Comm. Part. Differ. Equ., 10:10 (1985), 1117–1212 | DOI | MR | Zbl

[68] Eskin G., “Mixed initial-boundary value problem for second hyperbolic equations”, Comm. Part. Differ. Equ., 12:5 (1987), 503–587 | DOI | MR | Zbl

[69] Eskin G., “The wave equation in a wedge with general boundary conditions”, Comm. Part. Differ. Equ., 17:1–2 (1992), 99–160 | MR | Zbl

[70] Friedrichs K., “Symmetric hyperbolic linear differential equations”, Comm. Pure Appl. Math., 7:2 (1954), 345–392 | DOI | MR | Zbl

[71] Hadamard J., “Propriétés d'une équation linéaire aux dérivées partielles du quatrieme ordre”, Tohôku Math. J., 37 (1933), 133–150 | Zbl

[72] Hadamard J., “Equations aux dérivées partielles. Les conditions definies en général. Le cas hyperbolique”, Enseign. Math., 35 (1936), 5–42 | MR | Zbl

[73] Korzyuk V. I., “The Goursat problem for hyperbolic linear third order equations”, Math. Modelling Analysis, 6:2 (2001), 270–279 | MR | Zbl

[74] Korzyuk V. I., “Generalized-classical solution of the mixed problems for hyperbolic equations of the second order”, Analytic Methods of Analysis and Differential Equations, AMADE-2003, eds. A. A. Kilbas, S. V. Rogosin, Cambridge Sci. Publ., Cambridge, 2006, 133–154 | Zbl

[75] Korzyuk V. I., Cheb E. C., “Generalized-classical solution of the mixed problems for a hyperbolic equation of the second order”, Analytic Methods of Analysis and Differential Equations, AMADE-2006, eds. A. A. Kilbas, S. V. Rogosin, Cambridge Sci. Publ., Cambridge, 2008, 165–180 | MR

[76] Miyake M., Hashimoto Y., “Newton polygons and Gevrey indices for linear partial differential operators”, Nagoya Math. J., 128 (1992), 15–47 | MR | Zbl

[77] Papi G., “A Goursat problem for the wave equation”, Boll. Unione Mat. Ital. VI Ser. C Anal. Funz. Appl., 5:1 (1986), 139–157 | MR | Zbl

[78] Petrowsky I., “Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen”, Mat. sb., 2(44):5 (1937), 815–870 | Zbl

[79] Pini B., “Un problema di valori al contorno per un'equazione a derivate parziali del terzo ordine con parte principale di tipo composito”, Rend. Sem. Fac. Sci. Univ. Cagliari, 27 (1958), 114–135 | MR | Zbl

[80] Sakamoto R., “Mixed problems for hyperbolic equations”, Math. Kyoto Univ., 10 (1970), 49–373

[81] Sjöstrand O., “Sur une équation aux dérivées partielles du type composite”, Ark. Mat. Astron. Fys. A, 25 (1936), 1–11 | Zbl

[82] Sjöstrand O., “Sur une équation aux dérivées partielles du type composite. II”, Ark. Mat. Astron. Fys. A, 26:1 (1937), 1–10 | Zbl

[83] Thomée V., “Estimates of the Friedrichs–Levy type for a hyperbolic equation with three characteristics”, Math. Scand., 3 (1955), 115–123 | MR | Zbl

[84] Thomée V., “Estimates of the Friedrichs–Lewy type for mixed problems in the theory of linear hyperbolic differential equations in two independent variables”, Math. Scand., 5 (1957), 93–113 | MR | Zbl

[85] Thomée V., “Existence proofs for mixed problems for hyperbolic differential equations in two independent variables by means of the continuity method”, Math. Scand., 6 (1958), 5–32 | MR | Zbl