Second-order hyperbolic equations with strong characteristic degeneracy at the initial hypersurface

A. V. Deryabina

Sbornik. Mathematics, Tome 191 (2000) no. 4, pp. 503-527

Résumé

Equations of the following form are considered: \begin{equation} \psi^2(t,x)u_{tt}+\varphi(t,x)u_t-\sum_{i,j}\bigl(a^{ij}(t,x)u_{x_i}\bigr)_{x_j}+\sum_ib^i(t,x)u_{x_i}+c(t,x)u=f(t,x), \tag{1} \end{equation} where \begin{gather*} (t,x)\in H=(0,T)\times\mathbb R^n, \qquad \psi(t,x)\geqslant 0, \qquad \varphi(t,x)\geqslant0; \\ \sum_{i,j}a^{ij}(t,x)\xi_i\xi_j\geqslant0 \quad \forall\,(t,x)\in H, \quad \forall\,\xi=(\xi_1,\dots,\xi_n)\in\mathbb R^n. \end{gather*} In place of the Cauchy problem for (1), a problem without initial data but with constraints on the admissible growth of the solution as $t\to0$ and as $|x|\to\infty$ is discussed. The unique solubility of (1) in certain Sobolev-type weighted spaces is proved. The smoothness properties of generalized solutions are studied.

Bibliographie

[1] Baranovskii F. T., “Smeshannaya zadacha dlya giperbolicheskogo vyrozhdayuschegosya uravneniya”, Izv. vuzov. Ser. matem., 1960, no. 3, 30–42 | MR | Zbl

[2] Bitsadze A. V., Uravneniya smeshannogo tipa, Izd-vo AN SSSR, M., 1959

[3] Bryukhanov A. V., Kharakteristicheskaya zadacha Koshi dlya klassa giperbolicheskikh uravnenii, vyrozhdayuschikhsya v oblasti nachalnykh dannykh, 132, In-t matem. SO AN SSSR, Novosibirsk, 1987, 21–34 | MR

[4] Vainberg B. R., Mazya V. G., “Kharakteristicheskaya zadacha Koshi dlya giperbolicheskogo uravneniya”, Trudy sem. im. I. G. Petrovskogo, 7, Izd-vo MGU, M., 1981, 101–107 | MR

[5] Volevich L. R., Ivrii V. Ya., “Giperbolicheskie uravneniya”, kommentarii k rabotam [1],[3], Petrovskii I. G., Izbrannye trudy. Sistemy uravnenii s chastnymi proizvodnymi. Algebraicheskaya geometriya, Nauka, M., 1986, 395–418 | MR

[6] Volkodavov V. F., Repin O. A., “Ob odnoi zadache v polose dlya uravneniya giperbolicheskogo tipa s silnym vyrozhdeniem”, Differents. uravneniya, 22:8 (1986), 1442–1443 | MR | Zbl

[7] Vragov V. N., “Zadacha Koshi dlya nekotorykh klassov simmetricheskikh sistem pervogo poryadka i giperbolicheskikh uravnenii vtorogo poryadka”, Teoriya kubaturnykh formul i prilozheniya funktsionalnogo analiza k nekotorym zadacham matematicheskoi fiziki, In-t matem. SO AN SSSR, Novosibirsk, 1975, 44–47

[8] Glazatov S. N., “Zadacha Koshi dlya vyrozhdayuschegosya giperbolicheskogo uravneniya s proizvolnym kharakterom vyrozhdeniya”, Uravneniya neklassicheskogo tipa, In-t matem. SO AN SSSR, Novosibirsk, 1986, 44–47

[9] Tersenov S. A., “Ob odnom uravnenii giperbolicheskogo tipa, vyrozhdayuschemsya na granitse”, Dokl. AN SSSR, 129:2 (1959), 276–279 | MR | Zbl

[10] Baouendi M. S., Goulaouic C., “Cauchy problem with characteristic initial hypersurface”, Comm. Pure Appl. Math., 26:4 (1973), 455–475 | DOI | MR | Zbl

[11] Li Daqian, Yu Wenci, “Mixed-value problems for quasilinear hyperbolic equations with characteristic initial axis”, J. Fudan Univ. Natur. Sci., 26:1 (1987), 1–9 | MR | Zbl

[12] Bove A., Lewis J., Parenti C., Tahara H., “Cauchy problem for Fuchsian hyperbolic operators, II”, J. Fac. Sci. Univ. Tokyo. Sect. IA Math., 34:1 (1987), 127–157 | MR | Zbl

[13] Larkin N. A., “Zadacha Koshi dlya vyrozhdayuschegosya giperbolicheskogo uravneniya”, Differents. uravneniya, 9:1 (1973), 186–188 | MR | Zbl

[14] Malovichko V. A., “O kraevykh zadachakh dlya vyrozhdayuschikhsya i nevyrozhdayuschikhsya giperbolicheskikh sistem”, Differents. uravneniya, 25:11 (1989), 1977–1981 | MR | Zbl

[15] Nersesyan A. B., “O beskonechno differentsiruemykh resheniyakh zadachi Koshi dlya vyrozhdayuschikhsya giperbolicheskikh uravnenii”, Izv. AN Arm.SSR. Ser. matem., 4:3 (1969), 182–191 | MR

[16] Oganesyan G. R., “Nachalnaya zadacha dlya nekotorykh singulyarnykh giperbolicheskikh uravnenii i VKB-otsenki ee reshenii”, Differents. uravneniya, 25:6 (1989), 1062–1064 | MR | Zbl

[17] Oleinik O. A., “Zadacha Koshi i kraevaya zadacha dlya giperbolicheskikh uravnenii vtorogo poryadka, vyrozhdayuschikhsya v oblasti i na ee granitse”, Dokl. AN SSSR, 169:3 (1966), 525–528 | MR | Zbl

[18] Oleinik O. A., “O giperbolicheskikh uravneniyakh vtorogo poryadka, vyrozhdayuschikhsya vnutri oblasti i na ee granitse”, UMN, 24:2 (1969), 229–230 | MR | Zbl

[19] Oleinik O. A., Radkevich E. V., Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi, Itogi nauki. Matem. analiz 1969, 7, VINITI, M., 1971 | MR | Zbl

[20] Smirnov M. M., Vyrozhdayuschiesya giperbolicheskie uravneniya, Vysheishaya shkola, Minsk, 1977 | MR | Zbl

[21] Kalashnikov A. S., “Zadacha bez nachalnykh uslovii dlya lineinykh vyrozhdayuschikhsya giperbolicheskikh uravnenii vtorogo poryadka s beskonechnoi oblastyu zavisimosti”, Matem. sb., 88 (130):4 (8) (1972), 609–622 | MR | Zbl

[22] Deryabina A. V., “O zadache bez nachalnykh uslovii dlya silno vyrozhdayuschikhsya giperbolicheskikh uravnenii s rastuschimi koeffitsientami”, Vestn. MGU. Ser 1. Matem., mekh., 1987, no. 6, 71–73 | MR

[23] Deryabina A. V., “O rastuschikh resheniyakh silno vyrozhdayuschikhsya giperbolicheskikh sistem”, Matem. sb., 181:4 (1990), 447–463 | MR | Zbl

[24] Ladyzhenskaya O. A., Smeshannaya zadacha dlya giperbolicheskogo uravneniya, GITTL, M., 1953

[25] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 | MR