Geodesic
First Darboux Problem for Nonlinear Hyperbolic Equations of Second Order
Matematičeskie zametki, Tome 84 (2008) no. 5, pp. 693-712.

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We study the first Darboux problem for hyperbolic equations of second order with power nonlinearity. We consider the question of the existence and nonexistence of global solutions to this problem depending on the sign of the parameter before the nonlinear term and the degree of its nonlinearity. We also discuss the question of local solvability of the problem.
Keywords: first Darboux problem, nonlinear hyperbolic equation of second order, integral equation of Volterra type, Green–Hadamard function, Leray–Schauder theorem. 
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O. M. Dzhokhadze; S. S. Kharibegashvili. First Darboux Problem for Nonlinear Hyperbolic Equations of Second Order. Matematičeskie zametki, Tome 84 (2008) no. 5, pp. 693-712. http://geodesic.mathdoc.fr/item/MZM_2008_84_5_a5/

[1] A. V. Bitsadze, Nekotorye klassy uravnenii v chastnykh proizvodnykh, Nauka, M., 1981 | MR | Zbl

[2] F. John, “Blow-up of solutions of nonlinear wave equation in three space dimensions”, Manuscripta Math., 28:1–3 (1979), 235–268 | DOI | MR | Zbl

[3] F. John, “Blow-up for quasi-linear wave equations in three space dimensions”, Comm. Pure Appl. Math., 34:1 (1981), 29–51 | DOI | MR | Zbl

[4] F. John, S. Klainerman, “Almost global existence to nonlinear wave equations in three space dimensions”, Comm. Pure Appl. Math., 37:4 (1984), 443–455 | DOI | MR | Zbl

[5] T. Kato, “Blow-up of solutions of some nonlinear hyperbolic equations”, Comm. Pure Appl. Math., 33:4 (1980), 501–505 | DOI | MR | Zbl

[6] V. Georgiev, H. Lindblad, C. Sogge, “Weighted Strichartz estimates and global existence for semilinear wave equations”, Amer. J. Math., 119:6 (1977), 1291–1319 | DOI | MR | Zbl

[7] T. C. Sideris, “Nonexistence of global solutions to semilinear wave equations in high dimensions”, J. Differential Equations, 52:3 (1984), 378–406 | DOI | MR | Zbl

[8] L. Hörmander, Lectures on nonlinear hyperbolic differential equations, Math. Appl. (Berlin), 26, Springer-Verlag, Berlin, 1997 | MR | Zbl

[9] S. I. Pohozaev, L. Véron, “Blow-up results for nonlinear hyperbolic inequaties”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29:2 (2000), 393–420 | MR | Zbl

[10] E. Mitidieri, S. I. Pokhozhaev, Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001 | MR | Zbl

[11] G. Todorova, E. Vitillaro, “Blow-up for nonlinear dissipative wave equations in $\mathbb R^n$”, J. Math. Anal. Appl., 303:1 (2005), 242–257 | DOI | MR | Zbl

[12] E. Gursa, Kurs matematicheskogo analiza, t. 3, ch. 1: Beskonechno blizkie integraly. Uravneniya s chastnymi proizvodnymi, Gostekhizdat, M.–L., 1933 | MR | Zbl

[13] E. I. Moiseev, “O priblizhenii klassicheskogo resheniya zadachi Darbu gladkimi resheniyami”, Differents. uravneniya, 20:1 (1984), 73–87 | MR | Zbl

[14] E. I. Moiseev, Uravneniya smeshannogo tipa so spektralnym parametrom, Izd-vo MGU, M., 1988 | MR | Zbl

[15] S. Kharibegashvili, Goursat and Darboux Type Problems for Linear Hyperbolic Partial Differential Equations and Systems, Mem. Differential Equations Math. Phys., 4, Georgian Acad. Sci., Tbilisi, 1995 | MR | Zbl

[16] O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR | Zbl

[17] D. Khenri, Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1985 | MR | Zbl

[18] M. E. Lerner, “O kachestvennykh svoistvakh funktsii Rimana”, Differents. uravneniya, 27:12 (1991), 2106–2120 | MR | Zbl

[19] A. M. Nakhushev, Uravneniya matematicheskoi biologii, Vysshaya shkola, M., 1995 | Zbl

[20] M. M. Smirnov, Uravneniya smeshannogo tipa, Nauka, M., 1970 | MR | Zbl

[21] R. Narasimkhan, Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, Mir, M., 1971 | MR | Zbl

[22] D. Gilbarg, N. S. Trudinger, Ellipticheskie uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[23] V. A. Trenogin, Funktsionalnyi analiz, Nauka, M., 1993 | MR | Zbl

[24] V. I. Zhegalov, A. N. Mironov, Differentsialnye uravneniya so starshimi chastnymi proizvodnymi, Monografii po matematike, 5, Kazansk. matem. obsch-vo, Kazan, 2001 | Zbl

[25] R. Kurant, Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR | Zbl